![2 Chapter 1 I NTRODUCTION TO GRAPH MODELS
1 . 1 G RAPHS AND DIG RAPHS
We think of a graph as a set of points in a plane or in 3-space and a set of line
segments (possibly curved) , each of which either joins two points or joins a point to
itself.
a
Figure 1 .1 .1 Line drawings of a graph A and a graph B.
Graphs are highly versatile models for analyzing a wide range of practical problems in
which points and connections between them have some physical or conceptual interpre
tation. Placing such analysis on solid footing requires precise definitions, terminology,
and notation.
DEFINITION: A graph G = (V, E) is a mathematical structure consisting of two finite
sets V and E. The elements of V are called vertices (or nodes) , and the elements of
E are called edges. Each edge has a set of one or two vertices associated to it, which
are called its endpoints.
TERMINOLOGY: An edge is said to join its endpoints. A vertex joined by an edge to a
vertex v is said to be a neighbor of v.
DEFINITION: The (open) neighborhood of a vertex v in a graph G, denoted N(v), is
the set of all the neighbors of v. The closed neighborhood of v is given by N[v] =
N(v) U {v}.
NOTATION: When G is not the only graph under consideration, the notations VG and
EG (or V(G) and E(G)) are used for the vertex- and edge-sets of G, and the notations
NG(v) and NG[v] are used for the neighborhoods of v.
Example 1.1.1: The vertex- and edge-sets of graph A in Figure 1.1.1 are given by
VA = {p, q, r, s} and EA = {pq, pr, ps, rs, qs}
and the vertex- and edge-sets of graph B are given by
VB = {u, v, w} and EB = {a, b, c, d, f, g, h, k}
Notice that in graph A, we are able to denote each edge simply by juxtaposing its
endpoints, because those endpoints are unique to that edge. On the other hand, in
graph B, where some edges have the same set of endpoints, we use explicit names to
denote the edges.
Simple Graphs and General Graphs
In certain applications of graph theory and in some theoretical contexts, there are
frequent instances in which an edge joins a vertex to itself or two or more vertices have](https://image.slidesharecdn.com/4976566-250523070119-0d965102/85/Graph-Theory-And-Its-Applications-Second-Edition-2nd-Edition-Jonathan-L-Gross-25-320.jpg)
![4 Chapter 1 I NTRODUCTION TO GRAPH MODELS
DEFINITION: A multi-arc is a set of two or more arcs having the same tail and same
head. The arc multiplicity is the number of arcs within the multi-arc.
DEFINITION: A directed graph (or digraph) is a graph each of whose edges is directed.
DEFINITION: A digraph is simple if it has neither self-loops nor multi-arcs.
NOTATION: In a simple digraph, an arc from vertex u to vertex v may be denoted by uv
or by the ordered pair [u, v] .
Example 1.1.2: The digraph in Figure 1.1.2 is simple. Its arcs are uv , v u , and v w .
Figure 1 .1 .2 A simple digraph with a pair of oppositely directed arcs.
DEFINITION: A mixed graph (or partially directed graph) is a graph that has both
undirected and directed edges.
DEFINITION: The underlying graph of a directed or mixed graph G is the graph that
results from removing all the designations of head and tail from the directed edges of
G (i.e., deleting all the edge-directions) .
Example 1.1.3: The digraph D in Figure 1.1.3 has the graph G as its underlying
graph.
<:0w <:agw
Figure 1 .1 .3 A digraph and its underlying graph.
Simple and non-simple graphs and digraphs all commonly arise as models; the
numerous and varied examples in §1.3 illustrate the robustness of our comprehensive
graph model.
Formal Specification of Graphs and Digraphs
Except for the smallest graphs, line drawings are inadequate for describing a graph;
imagine trying to draw a graph to represent a telephone network for a small city. Since
many applications involve computations on graphs having hundreds, or even thousands,
of vertices, another, more formal kind of specification of a graph is often needed.
The specification must include (implicitly or explicitly) a function endpts that spec
ifies, for each edge, the subset of vertices on which that edge is incident (i.e., its endpoint
set) . In a simple graph, the juxtaposition notation for each edge implicitly specifies its
endpoints, making formal specification simpler for simple graphs than for general graphs.](https://image.slidesharecdn.com/4976566-250523070119-0d965102/85/Graph-Theory-And-Its-Applications-Second-Edition-2nd-Edition-Jonathan-L-Gross-27-320.jpg)
![8 Chapter 1 I NTRODUCTION TO GRAPH MODELS
DEFINITION: The degree sequence of a graph is the sequence formed by arranging the
vertex degrees in non-increasing order.
Example 1.1.9: Figure 1.1.9 shows a graph and its degree sequence.
] a
w
g
< 6, 6, 4, 1, 1, 0 >
v u w z y x
Figure 1 .1 .9 A graph and its degree sequence.
Although each graph has a unique degree sequence, two structurally different graphs
can have identical degree sequences.
Example 1.1.10: Figure 1.1.10 shows two different graphs, G and H, with the same
degree sequence.
Figure 1 .1 .1 0 Two graphs whose degree sequences are both (3, 3, 2, 2, 2, 2).
The following theorem shows that the degree sequence of a simple graph must have
at least two equal terms. This has an interesting interpretation in a sociological model
that appears in Section 1.3 (see Application 1.3.2) .
Proposition 1.1.1. A non-trivial simple graph G must have at least one pair of vertices
whose degrees are equal.
Proof: Suppose that the graph G has n vertices. Then there appear to be n possible
degree values, namely 0, . . . , n - 1. However, there cannot be both a vertex of degree
0 and a vertex of degree n - 1, since the presence of a vertex of degree 0 implies that
each of the remaining n - 1 vertices is adjacent to at most n - 2 other vertices. Hence,
the n vertices of G can realize at most n - 1 possible values for their degrees. Thus, the
pigeonhole principle implies that at least two of the n vertices have equal degree. <)
The work of Leonhard Euler (1707-1783) is regarded as the beginning of graph the
ory as a mathematical discipline. The following result of Euler establishes a fundamental
relationship between the vertices and edges of a graph.
Theorem 1.1.2 [Euler's Degree-Sum Theorem]. The sum of the degrees of the
vertices of a graph is twice the number of edges.
Proof: Each edge contributes two to the degree sum.
Corollary 1.1.3. In a graph, there is an even number of vertices having odd degree.
Proof: Consider separately, the sum of the degrees that are odd and the sum of those
that are even. The combined sum is even by Theorem 1.1.2, and since the sum of the
even degrees is even, the sum of the odd degrees must also be even. Hence, there must
be an even number of vertices of odd degree. <)](https://image.slidesharecdn.com/4976566-250523070119-0d965102/85/Graph-Theory-And-Its-Applications-Second-Edition-2nd-Edition-Jonathan-L-Gross-31-320.jpg)
![1 0 Chapter 1 I NTRODUCTION TO GRAPH MODELS
vertex Vk such that Vk is adjacent to Vs but not to Vt. Let G be the graph obtained from
G by replacing the edges v1vtand VsVk with the edges v1vs and VtVk (as shown in Figure
1.1.12). Then the degrees are all preserved and v8 E NG(v!) n {v2 , . . . , Vd1+d· Thus,
INa(v!)n {v2 , . . . , Vd1+dl = r + 1, which contradicts the choice of G and completes the
�� 0
- Av:
:t
�',
Vz v3 Vs '
...._
'
'- Vt
Vk
Figure 1 .1 .1 2 Switching adjacencies while preserving all degrees.
Corollary 1.1.7 [Havel (1955) and Hakimi (1961)]. A sequence (d1 , d2, . . . , dn) of
nonnegative integers such that d1 2:: d2 2:: ... 2:: dn is graphic if and only if the sequence
(d2 - 1, . . . , dd1+1 - 1, dd1+2 , . . . , dn) is graphic. 0 (Exercises)
Remark: Corollary 1.1.7 yields a recurs1ve algorithm that decides whether a non
increasing sequence is graphic.
Algorithm 1 .1 .1 : Recursive GraphicSequence((d1, d2, . . . , dn))
Input: a non-increasing sequence (d1 , d2, . . . , dn)·
Output: TRUE if the sequence is graphic; FALSE if it is not.
If d1 = 0
Return TRUE
Else
If dn < 0
Return FALSE
Else
Let (a1, a2, . . . , an-1) be a non-increasing permutation
of (d2 - 1, . . . , dd1+1 - 1, dd1+2 , . . . , dn)·
Return GraphicSequence((a1 , a2, . . . , an-1))
Remark: An iterative version of the algorithm GraphicSequence based on repeated
application ofCorollary 1.1.7 can also be written and is left as an exercise (see Exercises) .
Also, given a graphic sequence, the steps of the iterative version can be reversed to
construct a graph realizing the sequence. However many zeroes you get at the end of
the forward pass, start with that many isolated vertices. Then backtrack the algorithm,
adding a vertex each time. The following example illustrates these ideas.
Example 1.1.12: We start with the sequence (3, 3, 2, 2, 1, 1). Figure 1.1.13 illustrates
an iterative version of the algorithm GraphicSequence and then illustrates the back
tracking steps leading to a graph that realizes the original sequence. The hollow vertex
shown in each backtracking step is the new vertex added at that step.](https://image.slidesharecdn.com/4976566-250523070119-0d965102/85/Graph-Theory-And-Its-Applications-Second-Edition-2nd-Edition-Jonathan-L-Gross-33-320.jpg)
![1 4 Chapter 1 I NTRODUCTION TO GRAPH MODELS
1.1.33 Prove or disprove: There exists a simple graph with 13 vertices, 31 edges, three
1-valent vertices, and seven 4-valent vertices.
DEFINITION: Let G = (V, E) be a graph and let W � V. Then W is a vertex cover of
G if every edge is incident on at least one vertex in W. (See § 13.4.)
1.1.34 Find upper and lower bounds for the size of a minimum (smallest) vertex cover
of an n-vertex connected simple graph G. Then draw three 8-vertex graphs, one that
achieves the lower bound, one that achieves the upper bound, and one that achieves
neither.
1.1.35 Find a minimum vertex cover for the underlying graph of the mixed graph
shown in Figure 1.1.7.
1.1.36s The graph shown below represents a network of tunnels, where the edges are
sections of tunnel and the vertices are junction points. Suppose that a guard placed
at a junction can monitor every section of tunnel leaving it. Determine the minimum
number of guards and a placement for them so that every section of tunnel is monitored.
DEFINITION: An independent set of vertices in a graph G is a set of mutually non
adjacent vertices. (See §2.3 and §9.1.)
1.1.37 Find upper and lower bounds for the size of a maximum (largest) independent
set of vertices in an n-vertex connected graph. Then draw three 8-vertex graphs, one
that achieves the lower bound, one that achieves the upper bound, and one that achieves
neither.
1.1.38 Find a maximum independent set of vertices in the underlying graph of the
mixed graph shown in Figure 1.1.7.
1.1.39s Find a maximum independent set of vertices in the graph of Exercise 1.1.36.
DEFINITION: A matching in a graph G is a set of mutually non-adjacent edges in G.
(See §9.3 and §13.4.)
1.1.40 Find upper and lower bounds for the size of a maximum (largest) matching in
an n-vertex connected graph. Then draw three 8-vertex graphs, one that achieves the
lower bound, one that achieves the upper bound, and one that achieves neither.
1.1.41 Find a maximum matching in the underlying graph of the mixed graph shown
in Figure 1.1.7.
1.1.42s Find a maximum matching in the graph of Exercise 1.1.36.
DEFINITION: Let G = (V, E) be a graph and let W � V. Then W dominates G (or
is a dominating set of G) if every vertex in V is in W or is adjacent to at least one
vertex in W. That is, 1::/v E V, 3w E W, v E N [w] . (See §10.2.)
1.1.43 Find upper and lower bounds for the size of a minimum (smallest) dominating
set of an n-vertex connected graph. Then draw three 8-vertex graphs, one that achieves
the lower bound, one that achieves the upper bound, and one that achieves neither.](https://image.slidesharecdn.com/4976566-250523070119-0d965102/85/Graph-Theory-And-Its-Applications-Second-Edition-2nd-Edition-Jonathan-L-Gross-37-320.jpg)
![20 Chapter 1 I NTRODUCTION TO GRAPH MODELS
in use. If the graph is an interval graph, then there is an assignment of subintervals of
the interval (-1000, 1000) (by suitable scaling, if necessary) that is consistent with the
archeological find.
Figure 1 .2.1 5 A graph model of an archeological find.
Remark: Intersection graphs have been generalized to tolerance graphs, which are
discussed in §10.4.
Line Graphs
Line graphs are a special case of intersection graphs.
DEFINITION: The line graph L(G) of a graph G has a vertex for each edge of G, and
two vertices in L (G) are adjacent if and only if the corresponding edges in G have a
vertex in common.
Thus, the line graph L(G) is the intersection graph corresponding to the endpoint sets
of the edges of G.
Example 1.2.13: Figure 1.2.16 shows a graph G and its line graph L(G) .
a
[]Y. 'ktJ:
d
G L(G}
Figure 1 .2.1 6 A graph and its line graph.
EXERCISES for Section 1.2
1.2.1s Find the number of edges for each of the following graphs.
a. Kn b. Km,n
1.2.2 What is the maximum possible number of edges in a simple bipartite graph on
m vertices?
1.2.3 Draw the smallest possible non-bipartite graph.
1.2.4s Determine the values of n for which the given graph is bipartite.
1.2.5 Draw a 3-regular bipartite graph that is not K3,3.](https://image.slidesharecdn.com/4976566-250523070119-0d965102/85/Graph-Theory-And-Its-Applications-Second-Edition-2nd-Edition-Jonathan-L-Gross-43-320.jpg)
![Section 1 .2 Common Families of Graphs 21
In Exercises 1.2.6 and 1.2. 7, determine whether the given graph is bipartite. In each
case, give a vertex bipartition or explain why the graph is not bipartite.
X Z
[]
1.2.7
O
y
w z
u v
u v
1.2.8 Label the vertices of the cube graph in Figure 1.2.5 with 3-digit binary strings
so that the labels on adjacent vertices differ in exactly one digit.
1.2.9 For each of the platonic graphs, is it possible to trace a tour of all the vertices
by starting at one vertex, traveling only along edges, never revisiting a vertex, and
never lifting the pen off the paper? Is it possible to make the tour return to the starting
vertex?
1.2.10s Prove or disprove: There does not exist a 5-regular graph on 11 vertices.
DEFINITION: A tournament is a digraph whose underlying graph is a complete graph.
1.2.11 a. Draw all the 3-vertex tournaments whose vertices are u, v , x .
b. Determine the number of 4-vertex tournaments whose vertices are u, v , x , y .
1.2.12 Prove that every tournament has at most one vertex of indegree 0 and at most
one vertex of outdegree 0.
1.2.13s Suppose that n vertices v1 , v2 , . . . ,Vn are drawn in the plane. How many
different n-vertex tournaments can be drawn on those vertices?
1.2.14 Chartrand and Lesniak [ChLe04] define a pair of sequences of nonnegative
integers (a1, a2, . . . , ar) and (b1 , b2 , . . . , bt) to be bigraphical if there exists a bipartite
graph G with bipartition subsets U = {u1 , u2 , . . . , Ur} and W = {w1 , w2 , . . . , Wt} such
that deg(ui ) = ai , i = 1, 2, . . . , r, and deg(wi ) = bi , i = 1, 2, . . . , t. Prove that a pair
of non-increasing sequences of nonnegative integers (a1 , a2, . . . , ar) and (b1 , b2 , . . . , bt)
with r 2': 2, 0 < a1 ::; t, and 0 < a1 ::; r is bigraphical if and only if the pair (a2, . . . , ar)
and (b1 - 1, b2 - 1, . . . , ba, - 1, ba,+l , ba,+2 , . . . , bt) is bigraphical.
1.2.15
1.2.16
Find all the 4-vertex circulant graphs.
Show that each of the following graphs is a circulant graph.
a. �
I I I
' 0
1.2.17 State a necessary and sufficient condition on the positive integers n and k for
eire(n : k) to be the cycle graph Cn.
1.2.18 Find necessary and sufficient conditions on the positive integers n and k for
eire(n : k) to be the graph consisting of n/2 mutually non-adjacent edges.
1.2.19 Determine the size of a smallest dominating set (defined in §1.1 exercises) m
the graph indicated.
b. Km,n e. CLn
1.2.20 Determine the size of a smallest vertex cover (defined in §1.1 exercises) in the
graph indicated.
b. Km,n e. CLn](https://image.slidesharecdn.com/4976566-250523070119-0d965102/85/Graph-Theory-And-Its-Applications-Second-Edition-2nd-Edition-Jonathan-L-Gross-44-320.jpg)
![Section 1 .3 Graph Modeling Applications 25
Models Requiring Non-Simple Graphs
Application 1.3.7 Roadways Between States: If in the Geographic-Adjacency Ap
plication 1.3.3, each edge joining two vertices represented a road that crosses a border
between the corresponding two states, then the graph would be non-simple, since pairs
of bordering states have more than one road joining them.
Application 1.3.8 Chemical Molecules: The benzene molecule shown in Figure
1.3.7 has double bonds for some pairs of its atoms, so it is modeled by a non-simple
graph. Since each carbon atom has valence 4, corresponding to four electrons in its
outer shell, it is represented by a vertex of degree 4; and since each hydrogen atom has
one electron in its only shell, it is represented by a vertex of degree 1.
Figure 1 .3.7 The benzene molecule.
Models That Use Simple Digraphs
For each of the next series of applications, a link in one direction does not imply a
link in the opposite direction.
Application 1.3.9 Ecosystems: The feeding relationships among the plant and an
imal species of an ecosystem is called a food web and may be modeled by a simple
digraph. The food web for a Canadian willow forest is illustrated in Figure 1.3.8.t Each
species in the system is represented by a vertex, and a directed edge from vertex u to
vertex v means that the species corresponding to u feeds on the species corresponding
to v .
meadow leaf pussy
willow beetle willow
�--.....----.
Figure 1 .3.8 The food web in a Canadian willow forest.
t This illustration was adapted from [WiWa90] , p.69.](https://image.slidesharecdn.com/4976566-250523070119-0d965102/85/Graph-Theory-And-Its-Applications-Second-Edition-2nd-Edition-Jonathan-L-Gross-48-320.jpg)
Graph Theory And Its Applications Second Edition 2nd Edition Jonathan L Gross
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- 6. DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN GRAPH THEORY AND ITS APPLICATIONS SECOND EDITION
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- 10. DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN GRAPH THEORY AND ITS APPLICATIONS SECOND EDITION JONATHAN L. GROSS JAY YELLEN Boca Raton London New York
- 11. Published iu 2006 by Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Jonathan L. Gross and Jay Yellen Chapman & Hall/CRC is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 International Standard Book Number-10: 1-58488-505-X (Hardcover) International Standard Book Number-13: 978-1-58488-505-4 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.cornl) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Catalog record is available from the Library of Congress informa Taylor & Francis Group is the Academic Division ofT&F Infonna pic. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
- 12. PREFACE Interest in graphs and their applications continues to grow rapidly, largely due to the usefulness of graphs as models for computation and optimization. This text targets the need for a comprehensive approach to the theory, integrating a careful exposition of classical developments with emerging methods, models, and practical needs. It is suit able for classroom presentation at the introductory graduate or advanced undergraduate level, or for self-study and reference by working professionals. Graph theory has evolved as a collection of seemingly disparate topics. The intent of the authors is to present this material in a more cohesive framework, characteristic of mathematical areas with longer traditions, such as linear algebra and group theory. In the process, important techniques and analytic tools are transformed into a unified mathematical methodology. Emphasis throughout is conceptual, with more than 800 graph drawings included to strengthen intuition and more than 2000 exercises ranging from routine drill to more challenging problem solving. Applications and concrete examples are designed to stim ulate interest and to demonstrate the relevance of new concepts. Algorithms are presented in a concise format, shorn of the details of computer implementation. Computer science students will find numerous projects inviting them to convert algorithms to computer programs. Software design issues are addressed throughout the book in the form of computational notes, which can be skipped by those not interested in implementation. These design issues include the efficient use of computational resources, software reusability, and the user interface. Summary of Contents Chapters 1 through 6 concentrate on graph representation, basic properties, model ing, and applications. Graphical constructions here are concrete and primarily spatial. When necessary, we introduce abstractions in a supportive role. Chapters 7 and 8 present some of the underpinnings of topological graph theory. Chapter 7 is devoted to planarity and Kuratowski's theorem, and in Chapter 8, the scope of graph drawings is expanded in multiple directions, including the topological model for drawings, drawings on higher-order surfaces, and computer drawings. Chapter 9 is about graph colorings, including vertex- and edge-colorings, map-colorings, and the related topics of cliques, independence numbers, and graph factorization. Chapter 10 discusses graph measurement, including measurements of graph map pings. Chapter 11 provides a brief introduction to analytic graph theory, which com prises three ofthe most extensively developed branches ofgraph theory - Ramsey theory, extremal graphs, and random graphs. The material in Chapters 12 and 13, special di graph models and network flows, overlaps with various areas in operations research and computer science. Chapters 14 through 16, the most algebraic chapters, concern enumeration, speci fication by voltage graphs, and constructing nonplanar layouts. Most of the material in Chapters 1 through 6 assumes no prerequisite. However, some familiarity with topics typically found in an undergraduate course in discrete mathematics would be useful, and those topics appear briefly in the appendix (along
- 13. with some linear algebra, which is used in §4.6) . The appendix also provides a quick review (including permutation groups) for some of the more advanced topics in the later chapters. To the Instructor The book has ample material for a two-semester course, and a variety of one semester courses can be designed with different slants using various combinations of chapters. An introductory one-semester general course in graph theory would typically include most of the topics covered in the first nine chapters, except possibly parts of Chapter 8. However, some instructors of a fast-paced one-semester course might consider leaving various sections of earlier chapters for self-study by students. This would allow more time for selections from later chapters. The definitions and results from all of Chapter 1, most of Chapter 2, and from §3.1 are used throughout the text, and we recommend that they be covered in any course that uses the book. The remaining chapters are largely independent of each other, with several notable exceptions, as follows: • §3.2 is used in §4.1. • §4.1 and §4.2 are used in §5.4 and §12.5. • §4.5 is referred to in §5.3 and §6.1. • Parts of Chapter 5 are used in § 13.3. • Chapter 8 is used in § 16.1 and §16.2. • §15.1, §15.2, and §15.3 are used in §16.3, § 16.4, and §16.5. A course oriented toward operations research/optimization should include most or all of the material in Chapters 4 (spanning trees) , 5 (connectivity) , 6 (traversability) , 9 (colorings) , 1 2 (digraph models) , and 1 3 (flows) , along with various other sections of the instructor's choice, depending on the time available. A course emphasizing the role of data structures and algorithms might add to the above topics more material from Chapter 3 (trees) . A more algebraic and topological course in graph theory might replace some of these selections with parts or all of Chapter 7 (planarity) , Chapter 8 (graph drawings) , Chapter 14 (enumeration) , Chapter 15 (voltage graphs) , and Chapter 16 (graphs on general surfaces) . New Features in the Second Edition • Solutions and Hints. Each exercise marked with a superscripts has a solution or hint appearing in the back of the book. • Supplementary Exercises. In addition to the section exercises, each chapter now concludes with a section of supplementary exercises, which are intended to develop the problem-solving skills of students and test whether they can go beyond what has been explicitly taught. Most of these exercises were designed as examination questions for students at Columbia University. • Foreshadowing. The first three chapters now preview a number ofconcepts, mostly via the exercises, that are more fully developed in later chapters. This makes it easier to encourage students to take earlier excursions in research areas that may be of particular interest to the instructor.
- 14. New Material in the Second Edition We were gratified to see the first edition of this book used successfully at many col leges and universities in North America, Europe, Asia, and Oceania. Suggestions from instructors at these institutions, and from their students, led to several improvements in this second edition, including the addition of much new material. The inclusion of other new material was inspired by contributions to our Handbook of Graph Theory. Almost nothing has been deleted from the first edition, and well over 100 pages of new or expanded material have been added to the second edition. The following 9 sections are all new: §8.6 Geometric Drawings §9.4 Factorization §10.1 Distance in Graphs §10.2 Domination in Graphs §10.3 Bandwidth §10.4 Intersection Graphs §11.1 Ramsey Graph Theory §11.2 Extremal Graph Theory §11.3 Random Graphs Highlights of Changes in the Second Edition Now, at the start of Chapter 1, we introduce the notation uv for an edge between vertices u and v of a simple graph. Our move to this more convenient notation was motivated by comments from our colleagues who work primarily in simple graphs. Of course, we continue to use explicit edge names in contexts where multiple edges may be present. We have expanded Chapter 2 with increased discussion of isomorphism testing, automorphisms and symmetry, vertex and edge orbits, graph reconstruction, and some graph operations that appeared much later in the first edition. We revised and reorganized some of the material in Chapters 3 and 4 to create a smoother flow and to make the unifying treatment of the tree-growing algorithms more transparent. The main change in Chapter 5 is the inclusion in §5.2 of a short proof of the Whitney-Robbins characterization of 2-edge-connected graphs. With the new Chapter 7, Kuratowski's Theorem and its proof (which were in Chapter 9 of the first edition) can now be studied without having to first cover the more demanding topology material in Chapter 8. Chapter 7 now also includes crossing numbers and thickness (from Chapter 15 of the first edition) , which are more closely related to planarity than to higher order surfaces. Most ofthe content ofthe old Chapter 7 (in the first edition) was redistributed to other chapters. Chapter 9 on graph colorings is essentially the old Chapter 10 together with the new section on factorization (§9.4) . Chapters 10 and 11 are entirely new, except for the two sections on graph mappings (§ 10.5 and § 10.6) , which migrated from the old Chapter 7. Except for some reorganiza tion within each chapter, Chapter 12 is the old Chapter 11, and Chapter 13 is the old Chapter 12. The new Chapters 14, 15, and 16 were the old Chapters 13, 14, and 15, respectively. Chapter 14 (graphical enumeration) has several improved proofs, especially ofBurnside's Lemma. Chapter 16 (non-planar layouts) no longer includes the sections on crossing numbers and thickness, now in Chapter 7, or the section on generalizing planar drawings to higher-order surfaces, now in Chapter 8.
- 15. First Edition to Second Edition at a Glance The section migrations are shown in the figure on the left in the diagram below. For instance, §4.4 in the first edition became §3.7 in the second edition. Substantial chapter rearrangements. 20 ....2 � 2.1 , 30 7 ...3 J - 3 .7r 4.4-/ 4 a- , .. e4 / 7.1 7 d 7.4- ' I I ) ,.8.5 9 I 'y / ' I / 10.5, 10.6 I/ 'e10 15.1':15.2 156' ...16 1st Edition 2nd Edition Websites Intact chapter transfers 10 ...1 50 ...5 60 ... 6 100 ... g 110 ....12 120 .... 13 130 ... 14 140 ... 15 1st Edition 2nd Edition Suggestions and comments from readers are welcomed and may be sent to the authors' website at www.graphtheory.com, which, thanks mostly to the efforts of our colleague Dan Sanders and our webmaster Aaron Gross, also maintains extensive graph theory informational resources. The general website for CRC Press is www.crcpress.com. In advance, we thank our students, colleagues, and other readers for notifying us of any errors that they may find. As with the first edition, we will post the corrections to all known errors on our website. Acknowledgements Several readers of our manuscript at various stages offered many helpful suggestions regarding the mathematical content. In particular, we would like to thank Bob Brigham, Jianer Chen, Lynn Kiaer, Ward Klein, Ben Manvel, Buck McMorris, Ken Rosen, Greg Starling, Joe Straight, Tom Tucker, and Dav Zimak. We also thank Betsey Maupin for her proofreading and for her many stylistic suggestions. Special thanks to Ward Klein for his considerable assistance with proofreading the manuscript for both editions. Jonathan Gross and Jay Yellen
- 16. ABOUT THE AUTHORS Jonathan Gross is Professor of Computer Science at Colum bia University. His research in topology, graph theory, and cultural sociometry has earned him an Alfred P. Sloan Fellow ship, an IBM Postdoctoral Fellowship, and various research grants from the Office of Naval Research, the National Sci ence Foundation, and the Russell Sage Foundation. Professor Gross has created and delivered numerous software development short courses for Bell Laboratories and for IBM. These include mathematical methods for performance evalu ation at the advanced level and for developing reusable soft ware at a basic level. He has received several awards for out standing teaching at Columbia University, including the career Great Teacher Award from the Society of Columbia Graduates. His peak semester enrollment in his graph theory course at Columbia was 101 students. His previous books include Topological Graph Theory, coauthored with Thomas W. Tucker. Another previous book, Measuring Culture, coauthored with Steve Rayner, constructs network-theoretic tools for measuring sociological phenomena. Prior to Columbia University, Professor Gross was in the Mathematics Department at Princeton University. His undergraduate work was at M.I.T., and he wrote his Ph.D. thesis on 3-dimensional topology at Dartmouth College. Jay Yellen is Professor of Mathematics at Rollins College. He received his B.S. and M.S. in Mathematics at Polytechnic University of New York and did his doctoral work in finite group theory at Colorado State University. Dr. Yellen has had regular faculty appointments at Allegheny College, State University of New York at Fredonia, and Florida Institute of Technology, where he was Chair of Operations Research from 1995 to 1999. He has had visiting appointments at Emory University, Georgia Institute of Technology, and Columbia University. In addition to the Handbook of Graph Theory, which he co edited with Professor Gross, Professor Yellen has written manuscripts used at IBM for two courses in discrete mathematics within the Principles of Computer Science Se ries and has contributed two sections to the Handbook of Discrete and Combinatorial Mathematics. He also has designed and conducted several summer workshops on cre ative problem solving for secondary-school mathematics teachers, which were funded by the National Science Foundation and New York State. He has been a recipient of a Student's Choice Professor Award at Rollins College. Dr. Yellen has published research articles in character theory of finite groups, graph theory, power-system scheduling, and timetabling. His current research interests include graph theory, discrete optimization, and graph algorithms for software testing and course timetabling.
- 18. Jonathan dedicates this book to Alisa. Jay dedicates this book to the memory of his brother Marty.
- 20. CONTENTS Preface 1. INTRODUCTION to GRAPH MODELS 1.1 Graphs and Digraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Common Families of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Graph Modeling Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.4 Walks and Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.5 Paths, Cycles, and Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.6 Vertex and Edge Attributes: More Applications . . . . . . . . . . . . . . 48 1.7 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2. STRUCTURE and REPRESENTATION 2.1 Graph Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.2 Automorphisms and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.3 Subgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.4 Some Graph Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.5 Tests for Non-Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.6 Matrix Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.7 More Graph Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.8 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3. TREES 3.1 Characterizations and Properties of Trees . . . . . . . . . . . . . . . . . . . 3.2 Rooted Trees, Ordered Trees, and Binary Trees . . . . . . . . . . . . . 3.3 Binary-Tree Traversals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Binary-Search Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Huffman Trees and Optimal Prefix Codes . . . . . . . . . . . . . . . . . . . 3.6 Priority Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Counting Labeled Trees: Priifer Encoding . . . . . . . . . . . . . . . . . . . 3.8 Counting Binary Trees: Catalan Recursion . . . . . . . . . . . . . . . . . . 3.9 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. SPANNING TREES 116 124 132 137 141 146 151 156 158 160 4.1 Tree Growing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.2 Depth-First and Breadth-First Search . . . . . . . . . . . . . . . . . . . . . . . 171 4.3 Minimum Spanning Trees and Shortest Paths . . . . . . . . . . . . . . . 176 4.4 Applications of Depth-First Search . . . . . . . . . . . . . . . . . . . . . . . . . . 182 4.5 Cycles, Edge-Cuts, and Spanning Trees . . . . . . . . . . . . . . . . . . . . . 190 4.6 Graphs and Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 4.7 Matroids and the Greedy Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 207 4.8 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 1 57 115 163
- 21. 5. CONNECTIVITY 5.1 Vertex- and Edge-Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Constructing Reliable Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Max-Min Duality and Menger's Theorems . . . . . . . . . . . . . . . . . . . 5.4 Block Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. OPTIMAL GRAPH TRAVERSALS 6.1 Eulerian Trails and Tours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 DeBruijn Sequences and Postman Problems . . . . . . . . . . . . . . . . . 6.3 Hamiltonian Paths and Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Gray Codes and Traveling Salesman Problems . . . . . . . . . . . . . . . 6.5 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. PLANARITY AND KURATOWSKI'S THEOREM 7.1 Planar Drawings and Some Basic Surfaces . . . . . . . . . . . . . . . . . . . 7.2 Subdivision and Homeomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Extending Planar Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Kuratowski's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Algebraic Tests for Planarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Planarity Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Crossing Numbers and Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. DRAWING GRAPHS AND MAPS 8.1 The Topology of Low Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Higher-Order Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Mathematical Model for Drawing Graphs . . . . . . . . . . . . . . . . . . . 8.4 Regular Maps on a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 lmbeddings on Higher-Order Surfaces . . . . . . . . . . . . . . . . . . . . . . . 8.6 Geometric Drawings of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. GRAPH COLORINGS 9.1 Vertex-Colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Map-Colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Edge-Colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 218 223 231 241 244 245 247 248 252 267 273 282 283 285 286 292 297 304 311 324 327 331 334 337 338 341 346 349 354 361 365 366 371 372 386 393 407 411 413
- 22. 10. MEASUREMENT AND MAPPINGS 10.1 Distance in Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 10.2 Domination in Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 10.3 Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 10.4 Intersection Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 10.5 Linear Graph Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 10.6 Modeling Network Emulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 10.7 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 11. ANALYTIC GRAPH THEORY 11.1 Ramsey Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 11.2 Extremal Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 11.3 Random Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 11.4 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 12. SPECIAL DIGRAPH MODELS 12.1 Directed Paths and Mutual Reachability . . . . . . . . . . . . . . . . . . . 494 12.2 Digraphs as Models for Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 505 12.3 Tournaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 12.4 Project Scheduling and Critical Paths . . . . . . . . . . . . . . . . . . . . . . 516 12.5 Finding the Strong Components of a Digraph . . . . . . . . . . . . . . 523 12.6 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 13. NETWORK FLOWS and APPLICATIONS 13.1 Flows and Cuts in Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 13.2 Solving the Maximum-Flow Problem . . . . . . . . . . . . . . . . . . . . . . . 542 13.3 Flows and Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 13.4 Matchings, Transversals, and Vertex Covers . . . . . . . . . . . . . . . . 560 13.5 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 14. GRAPHICAL ENUMERATION 14.1 Automorphisms of Simple Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 578 14.2 Graph Colorings and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 14.3 Burnside's Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 14.4 Cycle-Index Polynomial of a Permutation Group . . . . . . . . . . . 595 14.5 More Counting, Including Simple Graphs . . . . . . . . . . . . . . . . . . 600 14.6 Polya-Burnside Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 14.7 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 417 469 493 533 577
- 23. 15. ALGEBRAIC SPECIFICATION of GRAPHS 15.1 Cyclic Voltages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 15.2 Cayley Graphs and Regular Voltages . . . . . . . . . . . . . . . . . . . . . . . 623 15.3 Permutation Voltages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 15.4 Symmetric Graphs and Parallel Architectures . . . . . . . . . . . . . . 637 15.5 Interconnection-Network Performance . . . . . . . . . . . . . . . . . . . . . . 644 15.6 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 16. NONPLANAR LAYOUTS 16.1 Representing lmbeddings by Rotations . . . . . . . . . . . . . . . . . . . . . 652 16.2 Genus Distribution of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 16.3 Voltage-Graph Specification of Graph Layouts . . . . . . . . . . . . . 664 16.4 Non-KVL Imbedded Voltage Graphs . . . . . . . . . . . . . . . . . . . . . . . 670 16.5 Heawood Map-Coloring Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 16.6 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 APPENDIX A.1 Logic Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Some Basic Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Algebraic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Algorithmic Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Supplementary Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BIBLIOGRAPHY B.1 General Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SOLUTIONS and HINTS INDEXES 1.1 Index of Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Index of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Index of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 General Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 683 686 687 692 694 695 697 757 759 761 767 613 651 681 695 709 757
- 24. Chapter 1 INTRODUCTION TO GRAPH MODELS 1 .1 Graphs and Digraphs 1 .2 Common Families of Graphs 1 .3 Graph Modeling Applications 1 .4 Walks and Distance 1 .5 Paths, Cycles, and Trees 1 .6 Vertex and Edge Attributes: More Applications INTRODUCTION Configurations of nodes and connections occur in a great diversity of applications. They may represent physical networks, such as electrical circuits, roadways, or organic molecules. And they are also used in representing less tangible interactions as might occur in ecosystems, sociological relationships, databases, or the flow of control in a computer program. Formally, such configurations are modeled by combinatorial structures called graphs, consisting of two sets called vertices and edges and an incidence relation be tween them. The vertices and edges may have additional attributes, such as color or weight, or anything else useful to a particular model. Graph models tend to fall into a handful of categories. For instance, the network of one-way streets of a city requires a model in which each edge is assigned a direction; two atoms in an organic molecule may have more than one bond between them; and a computer program is likely to have loop structures. These examples require graphs with directions on their edges, with multiple connections between vertices, or with connections from a vertex to itself. In the past these different types of graphs were often regarded as separate entities, each with its own set of definitions and properties. We have adopted a unified approach by introducing all of these various graphs at once. This allows us to establish properties that are shared by several classes of graphs without having to repeat arguments. More over, this broader perspective has an added bonus: it inspires computer representations that lead to the design of fully reusable software for applications. We begin by introducing the basic terminology needed to view graphs both as configurations in space and as combinatorial objects that lend themselves to computer representation. Pure and applied examples illustrate how different kinds of graphs arise as models.
- 25. 2 Chapter 1 I NTRODUCTION TO GRAPH MODELS 1 . 1 G RAPHS AND DIG RAPHS We think of a graph as a set of points in a plane or in 3-space and a set of line segments (possibly curved) , each of which either joins two points or joins a point to itself. a Figure 1 .1 .1 Line drawings of a graph A and a graph B. Graphs are highly versatile models for analyzing a wide range of practical problems in which points and connections between them have some physical or conceptual interpre tation. Placing such analysis on solid footing requires precise definitions, terminology, and notation. DEFINITION: A graph G = (V, E) is a mathematical structure consisting of two finite sets V and E. The elements of V are called vertices (or nodes) , and the elements of E are called edges. Each edge has a set of one or two vertices associated to it, which are called its endpoints. TERMINOLOGY: An edge is said to join its endpoints. A vertex joined by an edge to a vertex v is said to be a neighbor of v. DEFINITION: The (open) neighborhood of a vertex v in a graph G, denoted N(v), is the set of all the neighbors of v. The closed neighborhood of v is given by N[v] = N(v) U {v}. NOTATION: When G is not the only graph under consideration, the notations VG and EG (or V(G) and E(G)) are used for the vertex- and edge-sets of G, and the notations NG(v) and NG[v] are used for the neighborhoods of v. Example 1.1.1: The vertex- and edge-sets of graph A in Figure 1.1.1 are given by VA = {p, q, r, s} and EA = {pq, pr, ps, rs, qs} and the vertex- and edge-sets of graph B are given by VB = {u, v, w} and EB = {a, b, c, d, f, g, h, k} Notice that in graph A, we are able to denote each edge simply by juxtaposing its endpoints, because those endpoints are unique to that edge. On the other hand, in graph B, where some edges have the same set of endpoints, we use explicit names to denote the edges. Simple Graphs and General Graphs In certain applications of graph theory and in some theoretical contexts, there are frequent instances in which an edge joins a vertex to itself or two or more vertices have
- 26. Section 1 . 1 Graphs and Digraphs 3 the same set of endpoints. In other applications or theoretical contexts, such instances are absent. DEFINITION: A proper edge is an edge that joins two distinct vertices. DEFINITION: A self-loop is an edge that joins a single endpoint to itself.t DEFINITION: A multi-edge is a collection of two or more edges having identical end points. The edge multiplicity is the number of edges within the multi-edge. DEFINITION: A simple graph has neither self-loops nor multi-edges. DEFINITION: A loopless graph (or multi-graph) may have multi-edges but no self loops. DEFINITION: A (general) graph may have self-loops and/or multi-edges. Example 1.1.1, continued: Graph A in Figure 1.1.1 is simple. Graph B is general; the edges a, b, and k are self-loops, and the edge-sets {f, g, h} and {a, b} are multi-edges. TERMINOLOGY: When we use the term graph without a modifier, we mean a general graph. An exception to this convention occurs when an entire section concerns simple graphs only, in which case, we make an explicit declaration at the beginning of that section. TERMINOLOGY NOTE: Some authors use the term graph without a modifier to mean simple graph, and they use pseudograph to mean general graph. Null and Trivial Graphs DEFINITION: A null graph is a graph whose vertex- and edge-sets are empty. DEFINITION: A trivial graph is a graph consisting of one vertex and no edges. Edge Directions An edge between two vertices creates a connection in two opposite senses at once. Assigning a direction makes one of these senses forward and the other backward. In a line drawing, the choice of forward direction is indicated by placing an arrow on an edge. DEFINITION: A directed edge (or arc) is an edge, one of whose endpoints is designated as the tail, and whose other endpoint is designated as the head. TERMINOLOGY: An arc is said to be directed from its tail to its head. NOTATION: In a general digraph, the head and tail of an arc e may be denoted head(e) and tail(e), respectively. DEFINITION: Two arcs between a pair of vertices are said to be oppositely directed if they do not have the same head and tail. t We use the term "self-loop" instead of the more commonly used term "loop" because loop means something else in many applications.
- 27. 4 Chapter 1 I NTRODUCTION TO GRAPH MODELS DEFINITION: A multi-arc is a set of two or more arcs having the same tail and same head. The arc multiplicity is the number of arcs within the multi-arc. DEFINITION: A directed graph (or digraph) is a graph each of whose edges is directed. DEFINITION: A digraph is simple if it has neither self-loops nor multi-arcs. NOTATION: In a simple digraph, an arc from vertex u to vertex v may be denoted by uv or by the ordered pair [u, v] . Example 1.1.2: The digraph in Figure 1.1.2 is simple. Its arcs are uv , v u , and v w . Figure 1 .1 .2 A simple digraph with a pair of oppositely directed arcs. DEFINITION: A mixed graph (or partially directed graph) is a graph that has both undirected and directed edges. DEFINITION: The underlying graph of a directed or mixed graph G is the graph that results from removing all the designations of head and tail from the directed edges of G (i.e., deleting all the edge-directions) . Example 1.1.3: The digraph D in Figure 1.1.3 has the graph G as its underlying graph. <:0w <:agw Figure 1 .1 .3 A digraph and its underlying graph. Simple and non-simple graphs and digraphs all commonly arise as models; the numerous and varied examples in §1.3 illustrate the robustness of our comprehensive graph model. Formal Specification of Graphs and Digraphs Except for the smallest graphs, line drawings are inadequate for describing a graph; imagine trying to draw a graph to represent a telephone network for a small city. Since many applications involve computations on graphs having hundreds, or even thousands, of vertices, another, more formal kind of specification of a graph is often needed. The specification must include (implicitly or explicitly) a function endpts that spec ifies, for each edge, the subset of vertices on which that edge is incident (i.e., its endpoint set) . In a simple graph, the juxtaposition notation for each edge implicitly specifies its endpoints, making formal specification simpler for simple graphs than for general graphs.
- 28. Section 1 . 1 Graphs and Digraphs 5 DEFINITION: A formal specification of a simple graph is given by an adjacency table with a row for each vertex, containing the list of neighbors of that vertex. Example 1.1.4: Figure 1.1.4 shows a line drawing and a formal specification for a simple graph. rsr p : q r q : p s r : p s s : p q r s s r Figure 1 .1 .4 A simple graph and its formal specification. DEFINITION: A formal specification ofa general graph G = (V, E, endpts) consists of a list of its vertices, a list of its edges, and a two-row incidence table (specifying the endpts function) whose columns are indexed by the edges. The entries in the column corresponding to edge e are the endpoints of e. The same endpoint appears twice if e is a self-loop. (An isolated vertex will appear only in the vertex list.) Example 1.1.5: Figure 1.1.5 shows a line drawing and a formal specification for a general graph. a V = {u, v, w } and E = {a, b, c, d, J, g, h, k} g edge endpts w a b c d f g u u u w v v u u v u w w Figure 1 .1 .5 A general graph and its formal specification. h k w v v v DEFINITION: A formal specification of a general digraph or a mixed graph D = (V, E, endpts, head, tail) is obtained from the formal specification ofthe underlying graph by adding the functions head : Ea -+ Va and tail : Ea -+ Va, which designate the head vertex and tail vertex of each arc. One way to specify these designations in the incidence table is to mark in each column the endpoint that is the head of the corresponding arc. Remark: A mixed graph is specified by simply restricting the functions head and tail to a proper subset of Ea. In this case, a column of the incidence table that has no mark means that the corresponding edge is undirected. Example 1.1.6: Figure 1.1.6 gives the formal specification for the digraph shown, including the corresponding values of the functions head and tail. A superscript "h" is used to indicate the head vertex.
- 29. 6 edge endpts Chapter 1 I NTRODUCTION TO GRAPH MODELS a b c d f g h k head(a) = tail(a) = head(b) = tail(b) = head(d) = tail(c) = u; head(c) = head(h) = head(!) = tail(g) = head(k) = tail(k) = v; head(g) = tail(d) = tail(h) = tail(!) = w . Figure 1 .1 .6 A general digraph and its formal specification. Remark: Our approach treats a digraph as an augmented type of graph, where each edge e of a digraph is still associated with a subset endpts(e), but which now also includes a mark on one of the endpoints to specify the head of the directed edge. This viewpoint is partly motivated by its impact on computer implementations of graph algorithms (see the computational notes below) , but it has some advantages from a mathematical perspective as well. Regarding digraphs as augmented graphs makes it easier to view certain results that tend to be established separately for graphs and for digraphs as a single result that applies to both. Also, our formal incidence specification permits us to reverse the direction of an edge e at any time, just by reversing the values of head(e) and tail(e). This amounts to switching the h mark in the relevant column of the incidence table or reversing the arrowhead in the digraph drawing. COMPUTATIONAL NOTE 1 : These formal specifications for a graph and a digraph can easily be implemented with a variety of programmer-defined data structures, whatever is most appropriate to the application. A discussion of the comparative advantages and disadvantages of a few of the most common computer information structures for graphs and digraphs appears at the end of Chapter 2. COMPUTATIONAL NOTE 2: (A caution to software designers) From the perspective of object-oriented software design, the ordered-pair representation of arcs in a digraph treats digraphs as a different class of objects from graphs. This could seriously under mine software reuse. Large portions of computer code might have to be rewritten in order to adapt an algorithm that was originally designed for a digraph to work on an undirected graph. The ordered-pair representation could also prove awkward in implementing algorithms for which the graphs or digraphs are dynamic structures (i.e., they change during the algorithm) . Whenever the direction on a particular edge must be reversed, the associ ated ordered pair has to be deleted and replaced by its reverse. Even worse, if a directed edge is to become undirected, then an ordered pair must be replaced with an unordered pair. Similarly, the undirected and directed edges of a mixed graph would require two different types of objects. COMPUTATIONAL NOTE 3: For some applications (network layouts on a surface, for instance) , the direction of flow around a self-loop has practical importance, and distin guishing between the ends of a self-loop becomes necessary. This distinction is made in Chapters 8 and 16 but not elsewhere.
- 30. Section 1 . 1 Graphs and Digraphs Mathematical Modeling with Graphs 7 To bring the power of mathematics to bear on real-world problems, one must first model the problem mathematically. Graphs are remarkably versatile tools for modeling, and their wide-ranging versatility is a central focus throughout the text. Example 1.1.7: The mixed graph in Figure 1.1.7 is a model for a roadmap. The vertices represent landmarks, and the directed and undirected edges represent the one way and two-way streets, respectively. Gas Hardware Train Station Store Station Firehouse Grocery Town Hall Figure 1 .1 .7 Road-map of landmarks in a small town. Example 1.1.8: The digraph in Figure 1.1.8 represents the hierarchy of decision making within a company. This illustrates how, beyond physical networks, graphs and digraphs are used to model social relationships. Supervisors Staff Members Degree of a Vertex Figure 1 .1 .8 A corporate hierarchy. DEFINITION: Adjacent vertices are two vertices that are joined by an edge. DEFINITION: Adjacent edges are two edges that have an endpoint in common. DEFINITION: If vertex v is an endpoint of edge e, then v is said to be incident on e, and e is incident on v. DEFINITION: The degree (or valence) of a vertex v in a graph G, denoted deg(v) , 1s the number of proper edges incident on v plus twice the number of self-loops.t TERMINOLOGY: A vertex of degree d is also called a d-valent vertex. NOTATION: The smallest and largest degrees in a graph G are denoted Smin and Smax (or Smin (G) and Smax (G) when there is more than one graph under discussion) . Some authors use S instead of Smin and � instead of Smax . t Applications of graph theory to physical chemistry motivate the use of the term valence.
- 31. 8 Chapter 1 I NTRODUCTION TO GRAPH MODELS DEFINITION: The degree sequence of a graph is the sequence formed by arranging the vertex degrees in non-increasing order. Example 1.1.9: Figure 1.1.9 shows a graph and its degree sequence. ] a w g < 6, 6, 4, 1, 1, 0 > v u w z y x Figure 1 .1 .9 A graph and its degree sequence. Although each graph has a unique degree sequence, two structurally different graphs can have identical degree sequences. Example 1.1.10: Figure 1.1.10 shows two different graphs, G and H, with the same degree sequence. Figure 1 .1 .1 0 Two graphs whose degree sequences are both (3, 3, 2, 2, 2, 2). The following theorem shows that the degree sequence of a simple graph must have at least two equal terms. This has an interesting interpretation in a sociological model that appears in Section 1.3 (see Application 1.3.2) . Proposition 1.1.1. A non-trivial simple graph G must have at least one pair of vertices whose degrees are equal. Proof: Suppose that the graph G has n vertices. Then there appear to be n possible degree values, namely 0, . . . , n - 1. However, there cannot be both a vertex of degree 0 and a vertex of degree n - 1, since the presence of a vertex of degree 0 implies that each of the remaining n - 1 vertices is adjacent to at most n - 2 other vertices. Hence, the n vertices of G can realize at most n - 1 possible values for their degrees. Thus, the pigeonhole principle implies that at least two of the n vertices have equal degree. <) The work of Leonhard Euler (1707-1783) is regarded as the beginning of graph the ory as a mathematical discipline. The following result of Euler establishes a fundamental relationship between the vertices and edges of a graph. Theorem 1.1.2 [Euler's Degree-Sum Theorem]. The sum of the degrees of the vertices of a graph is twice the number of edges. Proof: Each edge contributes two to the degree sum. Corollary 1.1.3. In a graph, there is an even number of vertices having odd degree. Proof: Consider separately, the sum of the degrees that are odd and the sum of those that are even. The combined sum is even by Theorem 1.1.2, and since the sum of the even degrees is even, the sum of the odd degrees must also be even. Hence, there must be an even number of vertices of odd degree. <)
- 32. Section 1 . 1 Graphs and Digraphs 9 Corollary 1.1.4. The degree sequence of a graph is a finite, non-increasing sequence of nonnegative integers whose sum is even. <) Conversely, any non-increasing, nonnegative sequence of integers whose sum is even is the degree sequence of some graph. Theorem 1.1.5 prescribes how to construct such a graph. The following preliminary example illustrates the construction. Example 1.1.11: To construct a graph whose degree sequence is (5, 4, 3, 3, 2, 1, 0), start with seven isolated vertices v1,v2,...,v7. For the even-valued terms of the se quence, draw the appropriate number of self-loops on the corresponding vertices. Thus, v2 gets two self-loops, v5 gets one self-loop, and v7 remains isolated. For the four remaining odd-valued terms, group the corresponding vertices into any two pairs, for instance, v1,v3 and v4,v6. Then join each pair by a single edge and add to each vertex the appropriate number of self-loops. The resulting graph is shown in Figure 1.1.11. Figure 1 .1 .1 1 Constructing a graph with degree sequence (5, 4 , 3 , 3 , 2 , 1, 0). Theorem 1.1.5. Suppose that (d1,d2,. . . , dn ) is a sequence of nonnegative integers whose sum is even. Then there exists a graph with vertices v1,v2,. . . , Vn such that deg(vi) = di, for i = 1, . . . , n . Proof: Start with n isolated vertices v1,v2,. . . , Vn . For each i, if di is even, draw % self-loops on vertex vi, and if di is odd, draw d,;-1 self-loops. By Corollary 1.1.3, there is an even number of odd d;' s. Thus, the construction can be completed by grouping the vertices associated with the odd terms into pairs and then joining each pair by a single edge. <) Graphic Sequences The construction in Theorem 1.1.5 is straightforward but hinges on allowing the graph to be non-simple. A more interesting problem is determining when a sequence is the degree sequence of a simple graph. DEFINITION: A sequence (d1,d2,. . . , dn ) is said to be graphic if there is a permutation of it that is the degree sequence of some simple graph. Such a simple graph is said to realize the sequence. Theorem 1.1.6. Let (d1,d2,. . . , dn ) be a graphic sequence, with d1 2': d2 2': . . . 2': dn . Then there is a simple graph with vertex-set {v1,. . . , Vn } satisfying deg(vi) = di for i = 1, 2, . . . , n , such that v1 is adjacent to vertices v2,...,vd,+1. Proof: Among all simple graphs with vertex-set {v1,v2,. . . , Vn } and deg(vi) = di, i = 1, 2, . . . , n , let G be one for which r = lNG(v1)n {v2,...,vd,+dl is maximum. If r = d1, then the conclusion follows. If r < d1, then there exists a vertex vs, 2 ::; s ::; d1 + 1, such that v1 is not adjacent to V8 , and there exists a vertex Vt, t > d1+ 1 such that v1 is adjacent to Vt (since deg(vl) = dl). Moreover, since deg(vs) 2': deg(vt), there exists a
- 33. 1 0 Chapter 1 I NTRODUCTION TO GRAPH MODELS vertex Vk such that Vk is adjacent to Vs but not to Vt. Let G be the graph obtained from G by replacing the edges v1vtand VsVk with the edges v1vs and VtVk (as shown in Figure 1.1.12). Then the degrees are all preserved and v8 E NG(v!) n {v2 , . . . , Vd1+d· Thus, INa(v!)n {v2 , . . . , Vd1+dl = r + 1, which contradicts the choice of G and completes the �� 0 - Av: :t �', Vz v3 Vs ' ...._ ' '- Vt Vk Figure 1 .1 .1 2 Switching adjacencies while preserving all degrees. Corollary 1.1.7 [Havel (1955) and Hakimi (1961)]. A sequence (d1 , d2, . . . , dn) of nonnegative integers such that d1 2:: d2 2:: ... 2:: dn is graphic if and only if the sequence (d2 - 1, . . . , dd1+1 - 1, dd1+2 , . . . , dn) is graphic. 0 (Exercises) Remark: Corollary 1.1.7 yields a recurs1ve algorithm that decides whether a non increasing sequence is graphic. Algorithm 1 .1 .1 : Recursive GraphicSequence((d1, d2, . . . , dn)) Input: a non-increasing sequence (d1 , d2, . . . , dn)· Output: TRUE if the sequence is graphic; FALSE if it is not. If d1 = 0 Return TRUE Else If dn < 0 Return FALSE Else Let (a1, a2, . . . , an-1) be a non-increasing permutation of (d2 - 1, . . . , dd1+1 - 1, dd1+2 , . . . , dn)· Return GraphicSequence((a1 , a2, . . . , an-1)) Remark: An iterative version of the algorithm GraphicSequence based on repeated application ofCorollary 1.1.7 can also be written and is left as an exercise (see Exercises) . Also, given a graphic sequence, the steps of the iterative version can be reversed to construct a graph realizing the sequence. However many zeroes you get at the end of the forward pass, start with that many isolated vertices. Then backtrack the algorithm, adding a vertex each time. The following example illustrates these ideas. Example 1.1.12: We start with the sequence (3, 3, 2, 2, 1, 1). Figure 1.1.13 illustrates an iterative version of the algorithm GraphicSequence and then illustrates the back tracking steps leading to a graph that realizes the original sequence. The hollow vertex shown in each backtracking step is the new vertex added at that step.
- 34. Section 1 . 1 Graphs and Digraphs <3, 3, 2, 2, 1' 1 > t Cor. 1.1.7 <2, 1' 1' 1' 1 > t Cor. 1.1.7 <0, 0, 1, 1 > t permute <1, 1, 0, 0 > t Cor.1.1.7 <0, 0, 0 > 0 I t L I t • 1 • t • • • Figure 1 .1 .1 3 Testing and realizing the sequence (3, 3, 2, 2, 1, 1). lndegree and Outdegree in a Digraph The definition of vertex degree is slightly refined for digraphs. 1 1 DEFINITION: The indegree of a vertex v in a digraph is the number of arcs directed to v; the outdegree of vertex v is the number of arcs directed from v. Each self-loop at v counts one toward the indegree of v and one toward the outdegree. w vertex indegree outdegree u v w 3 4 1 3 2 3 Figure 1 .1 .1 4 The indegrees and outdegrees of the vertices of a digraph. The next theorem is the digraph version of Euler's Degree-Sum Theorem 1.1.2. Theorem 1.1.8. In a digraph, the sum of the indegrees and the sum of the outdegrees both equal the number of edges. Proof: Each directed edge e contributes one to the indegree at head(e) and one to the outdegree at tail(e). <) EXERCISES for Section 1.1 In Exercises 1.1.1 through 1.1.3, construct a line drawing, an incidence table, and the degree sequence of the graph with the given formal specification. 1.1.1s V = {u, w, x, z}; E = {e, f, g} endpts(e) = {w}; endpts(f) = {x, w}; endpts(g) = {x, z}
- 35. 1 2 Chapter 1 I NTRODUCTION TO GRAPH MODELS 1 . 1 .2 V = {u, v, x, y, z}; E = {a, b, c, d} endpts(a) = {u, v}; endpts(b) = {x, v}; endpts(c) = {u, v}; endpts(d) = {x} 1 . 1 .3 V = {u, v, x, y, z}; E = {e, f, g, h, k} endpts(e) = endpts(f) = {u, v}; endpts(g) = {x, z}; endpts(h) = endpts(k) = {y} In Exercises 1.1.4 through 1.1.61 construct a line drawing for the digraph or mixed graph with vertex-set V = {u, v, w , x, y}1 edge-set E = {e, f, g, h}1 and the given incidence table. edges e h endpts y X w h u X u 1 . 1 .5 edges e f g h endpts X v v v w h u uh uh 1 . 1 .6 edges e f g h endpts u xh v v w h u y u In Exercises 1.1. 7 through 1.1.91 give a formal specification for the given digraph. x� a y d b z c 1 . 1 .8 1 . 1 .9 v h In Exercises 1.1.10 through 1.1.121 give a formal specification for the underlying graph of the digraph indicated. 1 . 1 .10s The digraph of Exercise 1 . 1 .7. 1 . 1 .11 The digraph of Exercise 1 . 1 .8. 1 . 1 .12 The digraph of Exercise 1.1.9. 1 . 1 .13 Draw a graph with the given degree sequence. a. (8, 7, 3) b. (9, 8, 8, 6, 5, 3, 1)
- 36. Section 1 . 1 Graphs and Digraphs 1 . 1 .14 Draw a simple graph with the given degree sequence. a. (6, 4, 4, 3, 3, 2, 1 , 1) b. (5, 5, 5, 3, 3, 3, 3, 3) 1 3 For each of the number sequences in Exercises 1.1.15 through 1.1.18, either draw a simple graph that realizes it, or explain, without resorting to Corollary 1.1. 7 or Algorithm 1.1.1, why no such graph can exist. 1 . 1 .15s a. (2, 2, 1 , 0, 0) b. (4, 3, 2, 1 , 0) 1 . 1 .16 a. (4, 2, 2, 1 , 1) b. (2, 2, 2, 2) 1 . 1 .17 a. (4, 3, 2, 2, 1) b. (4, 3, 3, 3, 1) 1 . 1 .18 a. (4, 4, 4, 4, 3, 3, 3, 3) b. (3, 2, 2, 1 , 0) 1 . 1 .19 Apply Algorithm 1 . 1 . 1 to each of the following sequences to determine whether it is graphic. If the sequence is graphic, then draw a simple graph that realizes it. a. (7, 6, 6, 5, 4, 3, 2, 1) b. (5, 5, 5, 4, 2, 1 , 1, 1) c. (7, 7, 6, 5, 4, 4, 3, 2) d. (5, 5, 4, 4, 2, 2, 1, 1) 1 . 1 .20 Use Theorem 1.1.6 to prove Corollary 1.1.7. 1 . 1 .21 Write an iterative version of Algorithm 1 . 1 . 1 that applies Corollary 1 . 1 .7 re peatedly until a sequence of all zeros or a sequence with a negative term results. 1 . 1 .22s Given a group of nine people, is it possible for each person to shake hands with exactly three other people? 1 . 1 .23 Draw a graph whose degree sequence has no duplicate terms. 1 . 1 .24s What special property of a function must the endpts function have for a graph to have no multi-edges? 1 . 1 .25 Draw a digraph for each of the following indegree and outdegree sequences, such that the indegree and outdegree of each vertex occupy the same position in both sequences. a. m : (1 , 1 , 1) out: (1 , 1 , 1) b. in: (2, 1) out: (3, 0) DEFINITION: A pair ofsequences (a1 , a2, . . . , an ) and (b1, b2, . . . , bn ) is called digraphic ifthere exists a simple digraph with vertex-set {v1, v2, . . . , Vn } such that outdegree(vi) = ai and indegree(vi) = bi for i = 1 , 2, . . . , n. 1 . 1 .26 Determine whether the pair of sequences (3, 1, 1, 0) and (1 , 1, 1, 2) is digraphic. 1 . 1 .27 Establish a result like Corollary 1 . 1 .7 for a pair sequences to be digraphic. 1 . 1 .28 How many different degree sequences can be realized for a graph having three vertices and three edges? 1 . 1 .29 Given a list of three vertices and a list of seven edges, show that 37 different formal specifications for simple graphs are possible. 1 . 1 .30 Given a list of four vertices and a list of seven edges, show that G)56210 different formal specifications are possible if there are exactly two self-loops. 1 . 1 .31 Given a list of three vertices and a list of seven edges, how many different formal specifications are possible if exactly three of the edges are directed? 1 . 1 .32s Does there exist a simple graph with five vertices, such that every vertex is incident with at least one edge, but no two edges are adjacent?
- 37. 1 4 Chapter 1 I NTRODUCTION TO GRAPH MODELS 1.1.33 Prove or disprove: There exists a simple graph with 13 vertices, 31 edges, three 1-valent vertices, and seven 4-valent vertices. DEFINITION: Let G = (V, E) be a graph and let W � V. Then W is a vertex cover of G if every edge is incident on at least one vertex in W. (See § 13.4.) 1.1.34 Find upper and lower bounds for the size of a minimum (smallest) vertex cover of an n-vertex connected simple graph G. Then draw three 8-vertex graphs, one that achieves the lower bound, one that achieves the upper bound, and one that achieves neither. 1.1.35 Find a minimum vertex cover for the underlying graph of the mixed graph shown in Figure 1.1.7. 1.1.36s The graph shown below represents a network of tunnels, where the edges are sections of tunnel and the vertices are junction points. Suppose that a guard placed at a junction can monitor every section of tunnel leaving it. Determine the minimum number of guards and a placement for them so that every section of tunnel is monitored. DEFINITION: An independent set of vertices in a graph G is a set of mutually non adjacent vertices. (See §2.3 and §9.1.) 1.1.37 Find upper and lower bounds for the size of a maximum (largest) independent set of vertices in an n-vertex connected graph. Then draw three 8-vertex graphs, one that achieves the lower bound, one that achieves the upper bound, and one that achieves neither. 1.1.38 Find a maximum independent set of vertices in the underlying graph of the mixed graph shown in Figure 1.1.7. 1.1.39s Find a maximum independent set of vertices in the graph of Exercise 1.1.36. DEFINITION: A matching in a graph G is a set of mutually non-adjacent edges in G. (See §9.3 and §13.4.) 1.1.40 Find upper and lower bounds for the size of a maximum (largest) matching in an n-vertex connected graph. Then draw three 8-vertex graphs, one that achieves the lower bound, one that achieves the upper bound, and one that achieves neither. 1.1.41 Find a maximum matching in the underlying graph of the mixed graph shown in Figure 1.1.7. 1.1.42s Find a maximum matching in the graph of Exercise 1.1.36. DEFINITION: Let G = (V, E) be a graph and let W � V. Then W dominates G (or is a dominating set of G) if every vertex in V is in W or is adjacent to at least one vertex in W. That is, 1::/v E V, 3w E W, v E N [w] . (See §10.2.) 1.1.43 Find upper and lower bounds for the size of a minimum (smallest) dominating set of an n-vertex connected graph. Then draw three 8-vertex graphs, one that achieves the lower bound, one that achieves the upper bound, and one that achieves neither.
- 38. Section 1 .2 Common Families of Graphs 1 5 1 . 1 .44 Find a minimum dominating set for the underlying graph of the mixed graph shown in Figure 1 . 1 .7. 1 . 1 .45s Find a minimum dominating set for the graph of Exercise 1 . 1 .36. 1 .2 COMMON FAM I LI ES OF G RAPHS There is a multitude of standard examples that recur throughout graph theory. Complete Graphs DEFINITION: A complete graph is a simple graph such that every pair of vertices is joined by an edge. Any complete graph on n vertices is denoted Kn . Example 1.2.1: Complete graphs on one, two, three, four, and five vertices are shown in Figure 1 . 2 . 1 . Bipartite Graphs Figure 1 .2.1 The first five complete graphs. DEFINITION: A bipartite graph G is a graph whose vertex-set V can be partitioned into two subsets U and W, such that each edge of G has one endpoint in U and one endpoint in W. The pair U, W is called a (vertex) bipartition of G, and U and W are called the bipartition subsets. Example 1.2.2: Two bipartite graphs are shown in Figure 1 .2.2. The bipartition subsets are indicated by the solid and hollow vertices. D Figure 1 .2.2 Two bipartite graphs. Proposition 1.2.1. A bipartite graph cannot have any self-loops. Proof: This is an immediate consequence of the definition.
- 39. 1 6 Chapter 1 I NTRODUCTION TO GRAPH MODELS Example 1.2.3: The smallest possible simple graph that is not bipartite is the com plete graph K3, shown in Figure 1.2.3. Figure 1 .2.3 The smallest non-bipartite simple graph. DEFINITION: A complete bipartite graph is a simple bipartite graph such that every vertex in one of the bipartition subsets is joined to every vertex in the other bipartition subset. Any complete bipartite graph that has m vertices in one of its bipartition subsets and n vertices in the other is denoted Km,n. t Example 1.2.4: The complete bipartite graph K3,4 is shown in Figure 1.2.4. Figure 1 .2.4 The complete bipartite graph K3,4• Regular Graphs DEFINITION: A regular graph is a graph whose vertices all have equal degree. A k regular graph is a regular graph whose common degree is k. DEFINITION: The five regular polyhedra illustrated in Figure 1.2.5 are known as the platonic solids. Their vertex and edge configurations form regular graphs called the platonic graphs. Tetrahedron Cube Octahedron Dodecahedron Icosahedron Figure 1 .2.5 The five platonic graphs. t The sense in which f{m,n is a unique object is described in §2.1.
- 40. Section 1 .2 Common Families of Graphs 1 7 DEFINITION: The Petersen graph is the 3-regular graph represented by the line draw ing in Figure 1.2.6. Because it possesses a number of interesting graph-theoretic prop erties, the Petersen graph is frequently used both to illustrate established theorems and to test conjectures. Figure 1 .2.6 The Petersen graph. Example 1.2.5: The oxygen molecule 02, made up of two oxygen atoms linked by a double bond, can be represented by the 2-regular graph shown in Figure 1.2.7. 0==0 Figure 1 .2.7 A 2-regular graph representing the oxygen molecule 02• Bouquets and Dipoles One-vertex and two-vertex (non-simple) graphs often serve as building blocks for various interconnection networks, including certain parallel architectures (see Chap ter 15) . DEFINITION: A graph consisting of a single vertex with n self-loops is called a bouquet and is denoted En. Figure 1 .2.8 Bouquets B2 and B4• DEFINITION: A graph consisting of two vertices and n edges joining them is called a dipole and is denoted Dn. Example 1.2.6: The graph representation of the oxygen molecule in Figure 1.2.7 is an instance of the dipole D2. Figure 1.2.9 shows the dipoles D3 and D4. Figure 1 .2.9 Dipoles D3 and D4•
- 41. 1 8 Chapter 1 I NTRODUCTION TO GRAPH MODELS Path Graphs and Cycle Graphs DEFINITION: A path graph P is a simple graph with IVP I = IEP I + 1 that can be drawn so that all of its vertices and edges lie on a single straight line. A path graph with n vertices and n - 1 edges is denoted Pn . Example 1.2.7: Path graphs P2 and P4 are shown in Figure 1.2.10. Figure 1 .2.1 0 Path graphs P2 and P4• DEFINITION: A cycle graph is a single vertex with a self-loop or a simple graph C with IVc I = lEe I that can be drawn so that all of its vertices and edges lie on a single circle. An n-vertex cycle graph is denoted Cn . Example 1.2.8: The cycle graphs C1 , C2, and C4 are shown in Figure 1.2.11. Figure 1 .2.1 1 Cycle graphs C1 , C2, and C4• Remark: The terms path and cycle also refer to special sequences of vertices and edges within a graph and are defined in §1.4 and §1.5. Hypercubes and Circular Ladders DEFINITION: The hypercube graph Qn is the n-regular graph whose vertex-set is the set of bitstrings of length n, and such that there is an edge between two vertices if and only if they differ in exactly one bit. Example 1.2.9: The 8-vertex cube graph that appeared in Figure 1.2.5 is a hypercube graph Q3 (see Exercises) . DEFINITION: The circular ladder graph CLn is visualized as two concentric n-cycles in which each of the n pairs of corresponding vertices is joined by an edge. Example 1.2.10: The circular ladder graph CL4 is shown in Figure 1.2.12. Figure 1 .2.1 2 Circular ladder graph CL4•
- 42. Section 1 .2 Common Families of Graphs 1 9 Circulant Graphs DEFINITION : To the group of integers � = {0, 1, . . . , n - 1} under addition modulo n and a set S � { 1, . . . , n - 1} , we associate the circulant graph eire(n : S) whose vertex set is �n, such that two vertices i and j are adjacent if and only if there is a number s E S such that i + s = j mod n or j + s = i mod n. In this regard, the elements of the set S are called connections. NOTATI O N : It is often convenient to specify the connection set S = {s1 , . . . , Sr } without the braces, and to write eire(n : s1 , . . . , sr ). Example 1.2.11: Figure 1.2.13 shows three circulant graphs. 6 4 5 2 Figure 1 .2.1 3 The circulant graphs eire(5 : 1 , 2) , eire(6 : 1, 2) , and eire(8 : 1 , 4). Remark: Notice that circulant graphs are simple graphs. Circulant graphs are a special case of Cayley grophs, which are themselves derived from a special case of voltage grophs. Cayley graphs and voltage graphs, with applications, appear in Chapter 15. Intersection and Interval Graphs DEFINITI O N : A simple graph G with vertex-set Va = {v1 , v2 , . . . , vn} is an intersection graph if there exists a family of sets :F = {sl' s2' . . . ' Sn } such that vertex Vi is adjacent to Vj if and only if i :j:. j and si n Sj :j:. 0. DEFINITION : A simple graph is an interval graph if it is an intersection graph corre sponding to a family of intervals on the real line. Example 1.2.12: The graph in Figure 1 .2.14 is an interval graph for the following family of intervals: a ++ (1, 3) b t+ (2, 6) e t+ (5, 8) a�b dLJc d t+ (4, 7) Figure 1 .2.1 4 An interval graph. Application 1.2.1 Archeology: Suppose that a collection of artifacts was found at the site of a town known to have existed from 1000 BC to 1000 AD. In the graph shown below, the vertices correspond to the artifacts, and two vertices are adjacent if the corresponding artifacts appeared in the same grave. It is reasonable to assume that artifacts found in the same grave have overlapping time intervals during which they were
- 43. 20 Chapter 1 I NTRODUCTION TO GRAPH MODELS in use. If the graph is an interval graph, then there is an assignment of subintervals of the interval (-1000, 1000) (by suitable scaling, if necessary) that is consistent with the archeological find. Figure 1 .2.1 5 A graph model of an archeological find. Remark: Intersection graphs have been generalized to tolerance graphs, which are discussed in §10.4. Line Graphs Line graphs are a special case of intersection graphs. DEFINITION: The line graph L(G) of a graph G has a vertex for each edge of G, and two vertices in L (G) are adjacent if and only if the corresponding edges in G have a vertex in common. Thus, the line graph L(G) is the intersection graph corresponding to the endpoint sets of the edges of G. Example 1.2.13: Figure 1.2.16 shows a graph G and its line graph L(G) . a []Y. 'ktJ: d G L(G} Figure 1 .2.1 6 A graph and its line graph. EXERCISES for Section 1.2 1.2.1s Find the number of edges for each of the following graphs. a. Kn b. Km,n 1.2.2 What is the maximum possible number of edges in a simple bipartite graph on m vertices? 1.2.3 Draw the smallest possible non-bipartite graph. 1.2.4s Determine the values of n for which the given graph is bipartite. 1.2.5 Draw a 3-regular bipartite graph that is not K3,3.
- 44. Section 1 .2 Common Families of Graphs 21 In Exercises 1.2.6 and 1.2. 7, determine whether the given graph is bipartite. In each case, give a vertex bipartition or explain why the graph is not bipartite. X Z [] 1.2.7 O y w z u v u v 1.2.8 Label the vertices of the cube graph in Figure 1.2.5 with 3-digit binary strings so that the labels on adjacent vertices differ in exactly one digit. 1.2.9 For each of the platonic graphs, is it possible to trace a tour of all the vertices by starting at one vertex, traveling only along edges, never revisiting a vertex, and never lifting the pen off the paper? Is it possible to make the tour return to the starting vertex? 1.2.10s Prove or disprove: There does not exist a 5-regular graph on 11 vertices. DEFINITION: A tournament is a digraph whose underlying graph is a complete graph. 1.2.11 a. Draw all the 3-vertex tournaments whose vertices are u, v , x . b. Determine the number of 4-vertex tournaments whose vertices are u, v , x , y . 1.2.12 Prove that every tournament has at most one vertex of indegree 0 and at most one vertex of outdegree 0. 1.2.13s Suppose that n vertices v1 , v2 , . . . ,Vn are drawn in the plane. How many different n-vertex tournaments can be drawn on those vertices? 1.2.14 Chartrand and Lesniak [ChLe04] define a pair of sequences of nonnegative integers (a1, a2, . . . , ar) and (b1 , b2 , . . . , bt) to be bigraphical if there exists a bipartite graph G with bipartition subsets U = {u1 , u2 , . . . , Ur} and W = {w1 , w2 , . . . , Wt} such that deg(ui ) = ai , i = 1, 2, . . . , r, and deg(wi ) = bi , i = 1, 2, . . . , t. Prove that a pair of non-increasing sequences of nonnegative integers (a1 , a2, . . . , ar) and (b1 , b2 , . . . , bt) with r 2': 2, 0 < a1 ::; t, and 0 < a1 ::; r is bigraphical if and only if the pair (a2, . . . , ar) and (b1 - 1, b2 - 1, . . . , ba, - 1, ba,+l , ba,+2 , . . . , bt) is bigraphical. 1.2.15 1.2.16 Find all the 4-vertex circulant graphs. Show that each of the following graphs is a circulant graph. a. � I I I ' 0 1.2.17 State a necessary and sufficient condition on the positive integers n and k for eire(n : k) to be the cycle graph Cn. 1.2.18 Find necessary and sufficient conditions on the positive integers n and k for eire(n : k) to be the graph consisting of n/2 mutually non-adjacent edges. 1.2.19 Determine the size of a smallest dominating set (defined in §1.1 exercises) m the graph indicated. b. Km,n e. CLn 1.2.20 Determine the size of a smallest vertex cover (defined in §1.1 exercises) in the graph indicated. b. Km,n e. CLn
- 45. 22 Chapter 1 I NTRODUCTION TO GRAPH MODELS 1.2.21 Determine the size of a largest independent set of vertices (defined in § 1 . 1 exercises) in the graph indicated. b. Km,n e. CLn 1.2.22 Determine the size of a maximum matching (defined in § 1 . 1 exercises) in the graph indicated. a. Kn b. Km,n c. Cn d. Pn e. CLn 1.2.23s Show that the complete graph Kn is an interval graph for all n 2': 1 . 1.2.24 Draw the interval graph for the intervals (0, 2) , (3, 8) , ( 1 , 4) , (3, 4) , (2, 5), (7, 9). 1.2.25s Show that the graph modeling the archeological find in Application 1.2.1 is an interval graph by using a family of subintervals of the interval (-1000, 1000) . 1.2.26 Prove that the cycle graph Cn is not an interval graph for any n 2': 4. 1.2.27 Draw the intersection graph for the family of all subsets of the set {1, 2, 3}. 1.2.28 Prove that every simple graph is an intersection graph by describing how to construct a suitable family of sets. 1 .3 G RAPH MODELI NG APPLICATIONS Different kinds of graphs arise in modeling real-world problems. Sometimes, simple graphs are adequate; other times, non-simple graphs are needed. The analysis of some of the applications considered in this section is deferred to later chapters, where the necessary theoretical methods are fully developed. Models That Use Simple Graphs Application 1.3.1 Personnel-Assignment Problem: Suppose that a company re quires a number of different types of jobs, and suppose each employee is suited for some of these jobs, but not others. Assuming that each person can perform at most one job at a time, how should the jobs be assigned so that the maximum number ofjobs can be performed simultaneously? In the bipartite graph of Figure 1.3.1, the hollow vertices represent the employees, the solid vertices the jobs, and each edge represents a suitable job assignment. The bold edges represent a largest possible set of suitable assignments. Chapter 13 provides a fast algorithm to solve large instances of this classical problem in operations research. 'EcC=====�� J s Figure 1 .3.1 An optimal assignment of employees to jobs.
- 46. Section 1 .3 Graph Modeling Applications 23 Application 1.3.2 Sociological-Acquaintance Networks: In an acquaintance net work, the vertices represent persons, such as the students in a college class. An edge joining two vertices indicates that the corresponding pair of persons knew each other when the course began. The simple graph in Figure 1.3.2 shows a typical acquaintance network. Including the Socratic concept of self-knowledge would require the model to allow self-loops. For instance, a self-loop drawn at the vertex representing Slim might mean that she was "in touch" with herself. Lony Tooy� �-&( )/�' Susan Slim Figure 1 .3.2 An acquaintance network. By Proposition 1 . 1 . 1 , every group of two or more persons must contain at least two who know the same number of persons in the group. The acquaintance network of Figure 1 .3.2 has degree sequence (6, 4, 4, 4, 3, 3, 3, 3). Application 1.3.3 Geographic Adjacency: In the geographical model in Figure 1.3.3, each vertex represents a northeastern state, and each adjacency repre sents sharing a border. Maine Md. Del. Figure 1 .3.3 Geographic adjacency of the northeastern states. Application 1.3.4 Geometric Polyhedra: The vertex and edge configuration of any polyhedron in 3-space forms a simple graph, which topologists call its 1-skeleton. The 1-skeletons of the platonic solids, appearing in the previous section, are regular graphs. Figure 1 .3.4 shows a polyhedron whose 1-skeleton is not regular. Figure 1 .3.4 A non-regular !-skeleton of a polyhedron.
- 47. 24 Chapter 1 I NTRODUCTION TO GRAPH MODELS Application 1.3.5 Interconnection Networks for Parallel Architectures: Numerous processors can be linked together on a single chip for a multi-processor computer that can execute parallel algorithms. In a graph model for such an interconnection network, each vertex represents an individual processor, and each edge represents a direct link between two processors. Figure 1.3.5 illustrates the underlying graph structure of one such interconnection network, called a wrapped butterfly. Chapter 15 offers a glimpse of several parallel architectures, including the wrapped butterfly. Their specification illustrates some of the beautiful interplay between graph theory and abstract algebra. Figure 1 .3.5 A wrapped-butterfly-interconnection-network model. Application 1.3.6 Assigning Broadcasting Frequencies: When the radio transmit ters in a geographical region are assigned broadcasting frequencies, some pairs of trans mitters require different frequencies to avoid interference. A graph model can be used for the problem of minimizing the number of different frequencies assigned. Suppose that the seven radio transmitters, A,B,. . . , G, must be assigned frequencies. For simplicity, assume that if two transmitters are less than 100 miles apart, they must broadcast at different frequencies. Consider a graph whose vertices represent the transmitters, and whose edges indicate those pairs that are less than 100 miles apart. Figure 1.3.6 shows a table of distances for the seven transmitters and the corresponding graph on seven vertices. B c D E F G A 55 110 108 60 150 88 B 87 142 133 98 139 c c 77 91 85 93 G D 75 114 82 E 107 41 F 123 Figure 1 .3.6 A simple graph for a radio-frequency-assignment problem. The problem of assigning radio frequencies to avoid interference is equivalent to the problem of coloring the vertices of the graph so that adjacent vertices get different colors. The minimum number offrequencies will equal the minimum number of different colors required for such a coloring. This and several other graph-coloring problems and applications are discussed in Chapter 9.
- 48. Section 1 .3 Graph Modeling Applications 25 Models Requiring Non-Simple Graphs Application 1.3.7 Roadways Between States: If in the Geographic-Adjacency Ap plication 1.3.3, each edge joining two vertices represented a road that crosses a border between the corresponding two states, then the graph would be non-simple, since pairs of bordering states have more than one road joining them. Application 1.3.8 Chemical Molecules: The benzene molecule shown in Figure 1.3.7 has double bonds for some pairs of its atoms, so it is modeled by a non-simple graph. Since each carbon atom has valence 4, corresponding to four electrons in its outer shell, it is represented by a vertex of degree 4; and since each hydrogen atom has one electron in its only shell, it is represented by a vertex of degree 1. Figure 1 .3.7 The benzene molecule. Models That Use Simple Digraphs For each of the next series of applications, a link in one direction does not imply a link in the opposite direction. Application 1.3.9 Ecosystems: The feeding relationships among the plant and an imal species of an ecosystem is called a food web and may be modeled by a simple digraph. The food web for a Canadian willow forest is illustrated in Figure 1.3.8.t Each species in the system is represented by a vertex, and a directed edge from vertex u to vertex v means that the species corresponding to u feeds on the species corresponding to v . meadow leaf pussy willow beetle willow �--.....----. Figure 1 .3.8 The food web in a Canadian willow forest. t This illustration was adapted from [WiWa90] , p.69.
- 49. 26 Chapter 1 I NTRODUCTION TO GRAPH MODELS Application 1.3.10 Activity-Scheduling Networks: In large projects, often there are some tasks that cannot start until certain others are completed. Figure 1.3.9 shows a digraph model of the precedence relationships among some tasks for building a house. Vertices correspond to tasks. An arc from vertex u to vertex v means that task v cannot start until task u is completed. To simplify the drawing, arcs that are implied by transitivity are not drawn. This digraph is the cover diagram of a partial ordering of the tasks. A different model, in which the tasks are represented by the arcs of a digraph, is studied in Chapter 12. Activity 8 1 Foundation 2 Walls and ceilings 3 Roof 6 4 Electrical wiring 5 Windows 6 Siding 7 Paint interior 8 Paint exterior Figure 1 .3.9 An activity digraph for building a house. Application 1.3.11 Flow Diagrams for Computer Programs: A computer program is often designed as a collection of program blocks, with appropriate flow control. A digraph is a natural model for this decomposition. Each program block is associated with a vertex, and if control at the last instruction of block u can transfer to the first instruction of block v, then an arc is drawn from vertex u to vertex v. Computer flow diagrams do not usually have multi-arcs. Unless a single block is permitted both to change values of some variables and to retest those values, a flow diagram with a self-loop would mean that the program has an infinite loop. Models Requiring Non-Simple Digraphs Application 1.3.12 Markov Diagrams: Suppose that the inhabitants of some re mote area purchase only two brands of breakfast cereal, O's and W's. The consumption patterns of the two brands are encapsulated by the transition matrix shown in Figure 1.3.10. For instance, if someone just bought O's, there is a 0.4 chance that the person's next purchase will be W's and a 0.6 chance it will be O's. In a Markov process, the transition probabilityofgoing from one state to another depends only on the current state. Here, states "0" and "W" correspond to whether the most recent purchase was O's or W's, respectively. The digraph model for this Markov process, called a Markov diagram, is shown in Figure 1.3.10. Each arc is labeled with the transition probability of moving from the state at the tail vertex to the state at the head. Thus, the probabilities on the outgoing edges from each vertex must sum to 1. This Markov diagram is an example of a weighted graph (see § 1.6) . Other examples of Markov diagrams appear in Chapter 12 . .4 .6 .7 .3 O's W's O's .6 .4 W's .3 .7 Figure 1 .3.1 0 A Markov diagram and its transition matrix.
- 50. Section 1 .3 Graph Modeling Applications 27 Application 1.3.13 Lexical Scanners: The source code of a computer program may be regarded as a string of symbols. A lexical scanner must scan these symbols, one at a time, and recognize which symbols "go together" to form a syntactic token or lexeme. We now consider a single-purpose scanner whose task is to recognize whether an input string of characters is a valid identifier in the C programming language. Such a scanner is a special case of a finite-state recognizer and can be modeled by a labeled digraph, as in Figure 1.3.11. One vertex represents the start state, in effect before any symbols have been scanned. Another represents the accept state, in which the substring of symbols scanned so far forms a valid C identifier. The third vertex is the reject state, indicating that the substring has been discarded because it is not a valid C identifier. Each arc label tells what kind of symbols causes a transition from the tail state to the head state. If the final state after the input string is completely scanned is the accept state, then the string is a valid C identifier. letter any char except letter or digit any char letter or digit Figure 1 .3.1 1 Finite-state recognizer for identifiers. EXERCISES for Section 1.3 1.3.1s Solve the radio-frequency-assignment problem in Application 1.3.6 by deter mining the minimum number of colors needed to color the vertices of the associated graph. Argue why the graph cannot be colored with fewer colors. 1.3.2 What is wrong with a computer program having the following abstract flow pattern? 1.3.3s Referring to the Markov diagram in Application 1.3.12, suppose that someone just purchased a box of O's. What is the probability that his next three purchases are W's, O's, and then W's? 1.3.4 Modify the finite-state recognizer in Application 1.3.13 so that it accepts only those strings that begin with two letters and whose remaining characters, if any, are digits. (Hint: consider adding one or more new states.) 1.3.5s Which strings are accepted by the following finite-state recognizer? 0
- 51. 28 Chapter 1 I NTRODUCTION TO GRAPH MODELS In Exercises 1.3.6 through 1.3.13, design appropriate graph or digraph models and prob lems for the given situation. That is, specify the vertices and edges and any additional features. 1.3.68 Two-person rescue teams are being formed from a pool of n volunteers from several countries. The only requirement is that both members of a team must have a language in common. 1.3.7 Suppose that meetings must be scheduled for several committees. Two com mittees must be assigned different meeting times if there is at least one person on both committees. 1.3.8 Represent the "relative strength" of a finite collection of logical propositional forms. For example, the proposition p 1 q is at least as strong as p V q since the first implies the second (i.e., (p 1 q) ::::} (p V q) is a tautology). 1.3.9 Suppose there are three power generators to be built in three of the seven most populated cities of a certain country. The distances between each pair of cities is given in the table shown. One would like to situate the generators so that each city is within 50 miles of at least one generator. A B c D E F B c 80 110 40 D 15 45 65 E F 60 100 55 20 35 70 50 55 25 G 80 90 80 40 60 70 1.3.10 Suppose there are k machines and l jobs, and each machine can do only a subset of the jobs. 1) Draw a graph to model this situation. 2) Express in terms of your graph model, the problem of assigning jobs to machines so that the maximum number ofjobs can be performed simultaneously. 1.3.11 A bridge tournament for five teams is to be scheduled so that each team plays two other teams. 1.3.12 Let R be a binary relation on a set S. (Relations are discussed in Ap- pendix A.2.) a. Describe a digraph model for a binary relation on a finite set. b. Draw the digraph for the relation R on the set S = {1, 2, 3, 4, 5}, given by R = {(1, 2) , (2, 1) , (1, 1) , (1, 5) , (4, 5) , (3, 3) }. 1.3.138 Describe in graph-theory terms, the digraph properties corresponding to each of the following possible properties of binary relations. a. reflexive; b. symmetric; c. transitive; d. antisymmetric. 1 .4 WALKS AND DISTANCE Many applications call for graph models that can represent traversal and distance. For instance, the number of node-links traversed by an email message on its route from
- 52. Section 1 .4 Walks and Distance 29 sender to recipient is a form of distance. Beyond physical distance, another example is that a sequence of tasks in an activity-scheduling network forms a critical path if a delay in any one of the tasks would cause a delay in the overall project completion. This section and the following one clarify the notion of walk and related terminology. Walks and Directed Walks In proceeding continuously from a starting vertex to a destination vertex of a physical representation of a graph, one would alternately encounter vertices and edges. Accordingly, a walk in a graph is modeled by such a sequence. DEFINITION: In a graph G, a walk from vertex vo to vertex Vn is an alternating sequence ofvertices and edges, such that endpts(e;) = {v;_1, v;}, for i = 1, ..., n. If G is a digraph (or mixed graph), then W is a directed walk if each edge e; is directed from vertex v;_1 to vertex v;, i.e., tail(e;) = v;_1 and head(e;) = v;. In a simple graph, there is only one edge beween two consecutive vertices of a walk, so one could abbreviate the representation as a vertex sequence W = (vo, v1, . . . , vn) In a general graph, one might abbreviate the representation as an edge sequence from the starting vertex to the destination vertex TERMINOLOGY: A walk (or directed walk) from a vertex x to a vertex y is also called an x-y walk (or x-y directed walk) . DEFINITION: The length of a walk or directed walk is the number of edge-steps in the walk sequence. DEFINITION: A walk of length 0, i.e., with one vertex and no edges, is called a trivial walk. DEFINITION: A closed walk (or closed directed walk) is a walk (or directed walk) that begins and ends at the same vertex. An open walk (or open directed walk) begins and ends at different vertices. Example 1.4.1: In Figure 1.4.1 below, there is an open walk of length 6, < OH, PA, NY, VT, MA, NY, PA > that starts at Ohio and ends at Pennsylvania. Notice that the given walk is an inefficient route, since it contains two repeated vertices and retraces an edge. § 1.5 establishes the terminology necessary for distinguishing between walks that repeat vertices and/or edges and those that do not.
- 53. 30 MD DE Chapter 1 I NTRODUCTION TO GRAPH MODELS ME Figure 1 .4.1 Geographic adjacency of the northeastern states. Example 1.4.2: In the Markov diagram below (from Application 1.3.12) , the choice sequence of a cereal eater who buys O's, switches to W's, sticks with W's for two more boxes, and then switches back to O's is represented by the closed directed walk (0, w, w, w, 0) The product of the transition probabilities along a walk in any Markov diagram equals the probability that the process will follow that walk during an experimental trial. Thus, the probability that this walk occurs, when starting from O's equals .4 x .7 x .7 x .3 = 0.0588. .4 .6 .7 .3 O's W's O's .6 .4 W's .3 .7 Figure 1 .4.2 Markov process from Application 1.3.12. Example 1.4.3: In the lexical scanner of Application 1.3.13, the identifier counter12 would generate an open directed walk of length 9 from the start vertex to the accept vertex. DEFINITION: The concatenation of two walks W1 = (vo, e1,...,Vk-1,ek,VkJ and W2 = (vk,ek+1,Vk+1,ek+2,...,Vn-1,en,VnJ such that walk W2 begins where walk W1 ends, is the walk W1 0 W2 = (vo, e1,...,Vk-1,ek,Vk,ek+1,...,Vn-1,en,VnJ· Example 1.4.4: Figure 1.4.3 shows the concatenation of a walk of length 2 with a walk of length 3. v w1 = (u, e,v, f, x) w w2 = (x, k, y, l, z, m, w) u w1 0 w2 = (u, e, v, f, x, k, y, l, z, m, w) z Figure 1 .4.3 Concatenation of two walks. DEFINITION: A subwalk of a walk W = (vo, e1,v1,e2,...,Vn-1,en,VnJ is a subsequence of consecutive entries S = (Vj,ej+1,VJ+1,...,ek,Vk) such that 0 ::; j ::; k ::; n , that begins and ends at a vertex. Thus, the subwalk is itself a walk. Example 1.4.5: In Figure 1.4.4, the closed directed walk (v, x, y, z, v) is a subwalk of the open directed walk (u, v, x, y, z, v, w, t).
- 54. Section 1 .4 Walks and Distance 31 y Figure 1 .4.4 Distance DEFINITION: The distance d(s, t) from a vertex s to a vertex t in a graph G is the length of a shortest s-t walk if one exists; otherwise, d(s, t) = oo. For digraphs, the directed distance d(s, t) is the length of a shortest directed walk from s to t. Example 1.4.6: In Figure 1.4.5, the distance from West Virginia to Maine is five. That is, starting in West Virginia, one cannot get to Maine without crossing at least five state borders. ME MD DE Figure 1 .4.5 Geographic adjacency of the northeastern states. A shortest walk (or directed walk) contains no repeated vertices or edges (see Exercises) . It is instructive to think about how one might find a shortest walk. Ad hoc approaches are adequate for small graphs, but a systematic algorithm is essential for larger graphs (see §4.3) . Figure 1 .4.6 How might you find a shortest walk from s to t in this graph? Eccentricity, Diameter, and Radius DEFINITION: The eccentricityofa vertex v in a graph G, denoted ecc(v) , is the distance from v to a vertex farthest from v. That is, ecc(v) = max{d(v, x)} x E Vo
- 55. Exploring the Variety of Random Documents with Different Content
- 56. § 12. But consider farther, not only to what, but by what, is the revelation. By sight? or word? If by sight, then to eyes which see justly. Otherwise, no sight would be revelation. So far, then, as your sight is just, it is the image of God’s sight. If by words,—how do you know their meanings? Here is a short piece of precious word revelation, for instance. “God is love.” Love! yes. But what is that? The revelation does not tell you that, I think. Look into the mirror, and you will see. Out of your own heart you may know what love is. In no other possible way,—by no other help or sign. All the words and sounds ever uttered, all the revelations of cloud, or flame, or crystal, are utterly powerless. They cannot tell you, in the smallest point, what love means. Only the broken mirror can. § 13. Here is more revelation. “God is just!” Just! What is that? The revelation cannot help you to discover. You say it is dealing equitably or equally. But how do you discern the equality? Not by inequality of mind; not by a mind incapable of weighing, judging, or distributing. If the lengths seem unequal in the broken mirror, for you they are unequal; but if they seem equal, then the mirror is true. So far as you recognize equality, and your conscience tells you what is just, so far your mind is the image of God’s: and so far as you do not discern this nature of justice or equality, the words “God is just” bring no revelation to you. § 14. “But His thoughts are not as our thoughts.” No: the sea is not as the standing pool by the wayside. Yet when the breeze crisps the pool, you may see the image of the breakers, and a likeness of the foam. Nay, in some sort, the same foam. If the sea is for ever invisible to you, something you may learn of it from the pool. Nothing, assuredly, any otherwise. “But this poor miserable Me! Is this, then, all the book I have got to read about God in?” Yes, truly so. No other book, nor fragment of book, than that, will you ever find;—no velvet-bound missal, nor frankincensed
- 57. manuscript;—nothing hieroglyphic nor cuneiform; papyrus and pyramid are alike silent on this matter;—nothing in the clouds above, nor in the earth beneath. That flesh-bound volume is the only revelation that is, that was, or that can be. In that is the image of God painted; in that is the law of God written; in that is the promise of God revealed. Know thyself; for through thyself only thou canst know God. § 15. Through the glass, darkly. But, except through the glass, in nowise. A tremulous crystal, waved as water, poured out upon the ground;— you may defile it, despise it, pollute it at your pleasure, and at your peril; for on the peace of those weak waves must all the heaven you shall ever gain be first seen; and through such purity as you can win for those dark waves, must all the light of the risen Sun of righteousness be bent down, by faint refraction. Cleanse them, and calm them, as you love your life. Therefore it is that all the power of nature depends on subjection to the human soul. Man is the sun of the world; more than the real sun. The fire of his wonderful heart is the only light and heat worth gauge or measure. Where he is, are the tropics; where he is not, the ice-world. 1 I have been embarrassed in assigning the names to these orders of art, the term “Contemplative” belonging in justice nearly as much to the romantic and pastoral conception as to the modern landscape. I intended, originally, to call the four schools—Romantic, Classic, Georgic, and Theoretic—which would have been more accurate; and more consistent with the nomenclature of the second volume; but would not have been pleasant in sound, nor to the general reader, very clear in sense.
- 58. CHAPTER II. THE LANCE OF PALLAS. § 1. It might be thought that the tenor of the preceding chapter was in some sort adverse to my repeated statement that all great art is the expression of man’s delight in God’s work, not in his own. But observe, he is not himself his own work: he is himself precisely the most wonderful piece of God’s workmanship extant. In this best piece not only he is bound to take delight, but cannot, in a right state of thought, take delight in anything else, otherwise than through himself. Through himself, however, as the sun of creation, not as the creation. In himself, as the light of the world.1 Not as being the world. Let him stand in his due relation to other creatures, and to inanimate things—know them all and love them, as made for him, and he for them;—and he becomes himself the greatest and holiest of them. But let him cast off this relation, despise and forget the less creation around him, and instead of being the light of the world, he is as a sun in space—a fiery ball, spotted with storm. § 2. All the diseases of mind leading to fatalest ruin consist primarily in this isolation. They are the concentration of man upon himself, whether his heavenly interests or his worldly interests, matters not; it is the being his own interests which makes the regard of them so mortal. Every form of asceticism on one side, of sensualism on the other, is an isolation of his
- 59. soul or of his body; the fixing his thoughts upon them alone: while every healthy state of nations and of individual minds consists in the unselfish presence of the human spirit everywhere, energizing over all things; speaking and living through all things. § 3. Man being thus the crowning and ruling work of God, it will follow that all his best art must have something to tell about himself, as the soul of things, and ruler of creatures. It must also make this reference to himself under a true conception of his own nature. Therefore all art which involves no reference to man is inferior or nugatory. And all art which involves misconception of man, or base thought of him, is in that degree false, and base. Now the basest thought possible concerning him is, that he has no spiritual nature; and the foolishest misunderstanding of him possible is, that he has or should have, no animal nature. For his nature is nobly animal, nobly spiritual—coherently and irrevocably so; neither part of it may, but at its peril, expel, despise, or defy the other. All great art confesses and worships both. § 4. The art which, since the writings of Rio and Lord Lindsay, is specially known as “Christian,” erred by pride in its denial of the animal nature of man;—and, in connection with all monkish and fanatical forms of religion, by looking always to another world instead of this. It wasted its strength in visions, and was therefore swept away, notwithstanding all its good and glory, by the strong truth of the naturalist art of the sixteenth century. But that naturalist art erred on the other side; denied at last the spiritual nature of man, and perished in corruption. A contemplative reaction is taking place in modern times, out of which it may be hoped a new spiritual art may be developed. The first school of landscape, named, in the foregoing chapter, the Heroic, is that of the noble naturalists. The second (Classical), and third (Pastoral), belong to
- 60. the time of sensual decline. The fourth (Contemplative) is that of modern revival. § 5. But why, the reader will ask, is no place given in this scheme to the “Christian” or spiritual art which preceded the naturalists? Because all landscape belonging to that art is subordinate, and in one essential principle false. It is subordinate, because intended only to exalt the conception of saintly or Divine presence:—rather therefore to be considered as a landscape decoration or type, than an effort to paint nature. If I included it in my list of schools, I should have to go still farther back, and include with it the conventional and illustrative landscape of the Greeks and Egyptians. § 6. But also it cannot constitute a real school, because its first assumption is false, namely, that the natural world can be represented without the element of death. The real schools of landscape are primarily distinguished from the preceding unreal ones by their introduction of this element. They are not at first in any sort the worthier for it. But they are more true, and capable, therefore, in the issue, of becoming worthier. It will be a hard piece of work for us to think this rightly out, but it must be done. § 7. Perhaps an accurate analysis of the schools of art of all time might show us that when the immortality of the soul was practically and completely believed, the elements of decay, danger, and grief in visible things were always disregarded. However this may be, it is assuredly so in the early Christian schools. The ideas of danger or decay seem not merely repugnant, but inconceivable to them; the expression of immortality and perpetuity is alone possible. I do not mean that they take no note of the absolute fact of corruption. This fact the early painters often compel themselves to look fuller in the front than any other men: as in the way they usually paint the Deluge (the raven feeding on the
- 61. bodies), and in all the various triumphs and processions of the Power of Death, which formed one great chapter of religious teaching and painting, from Orcagna’s time to the close of the Purist epoch. But I mean that this external fact of corruption is separated in their minds from the main conditions of their work; and its horror enters no more into their general treatment of landscape than the fear of murder or martyrdom, both of which they had nevertheless continually to represent. None of these things appeared to them as affecting the general dealings of the Deity with His world. Death, pain, and decay were simply momentary accidents in the course of immortality, which never ought to exercise any depressing influence over the hearts of men, or in the life of Nature. God, in intense life, peace, and helping power, was always and everywhere. Human bodies, at one time or another, had indeed to be made dust of, and raised from it; and this becoming dust was hurtful and humiliating, but not in the least melancholy, nor, in any very high degree, important; except to thoughtless persons, who needed sometimes to be reminded of it, and whom, not at all fearing the things much himself, the painter accordingly did remind of it, somewhat sharply. § 8. A similar condition of mind seems to have been attained, not unfrequently, in modern times, by persons whom either narrowness of circumstance or education, or vigorous moral efforts have guarded from the troubling of the world, so as to give them firm and childlike trust in the power and presence of God, together with peace of conscience, and a belief in the passing of all evil into some form of good. It is impossible that a person thus disciplined should feel, in any of its more acute phases, the sorrow for any of the phenomena of nature, or terror in any material danger which would occur to another. The absence of personal fear, the consciousness of security as great in the midst of pestilence and storm, as amidst beds of flowers on a summer morning, and the certainty that whatever appeared evil, or was assuredly painful, must eventually issue in a far greater and enduring good—this general feeling and conviction, I say, would gradually lull, and at last put to entire rest, the
- 62. physical sensations of grief and fear; so that the man would look upon danger without dread,—accept pain without lamentation. § 9. It may perhaps be thought that this is a very high and right state of mind. Unfortunately, it appears that the attainment of it is never possible without inducing some form of intellectual weakness. No painter belonging to the purest religious schools ever mastered his art. Perugino nearly did so; but it was because he was more rational— more a man of the world—than the rest. No literature exists of a high class produced by minds in the pure religious temper. On the contrary, a great deal of literature exists, produced by persons in that temper, which is markedly, and very far, below average literary work. § 10. The reason of this I believe to be, that the right faith of man is not intended to give him repose, but to enable him to do his work. It is not intended that he should look away from the place he lives in now, and cheer himself with thoughts of the place he is to live in next, but that he should look stoutly into this world, in faith that if he does his work thoroughly here, some good to others or himself, with which, however, he is not at present concerned, will come of it hereafter. And this kind of brave, but not very hopeful or cheerful faith, I perceive to be always rewarded by clear practical success and splendid intellectual power; while the faith which dwells on the future fades away into rosy mist, and emptiness of musical air. That result indeed follows naturally enough on its habit of assuming that things must be right, or must come right, when, probably, the fact is, that so far as we are concerned, they are entirely wrong; and going wrong: and also on its weak and false way of looking on what these religious persons call “the bright side of things,” that is to say, on one side of them only, when God has given them two sides, and intended us to see both.
- 63. § 11. I was reading but the other day, in a book by a zealous, useful, and able Scotch clergyman, one of these rhapsodies, in which he described a scene in the Highlands to show (he said) the goodness of God. In this Highland scene there was nothing but sunshine, and fresh breezes, and bleating lambs, and clean tartans, and all manner of pleasantness. Now a Highland scene is, beyond dispute, pleasant enough in its own way; but, looked close at, has its shadows. Here, for instance, is the very fact of one, as pretty as I can remember—having seen many. It is a little valley of soft turf, enclosed in its narrow oval by jutting rocks and broad flakes of nodding fern. From one side of it to the other winds, serpentine, a clear brown stream, drooping into quicker ripple as it reaches the end of the oval field, and then, first islanding a purple and white rock with an amber pool, it dashes away into a narrow fall of foam under a thicket of mountain ash and alder. The autumn sun, low but clear, shines on the scarlet ash-berries and on the golden birch-leaves, which, fallen here and there, when the breeze has not caught them, rest quiet in the crannies of the purple rock. Beside the rock, in the hollow under the thicket, the carcass of a ewe, drowned in the last flood, lies nearly bare to the bone, its white ribs protruding through the skin, raven- torn; and the rags of its wool still flickering from the branches that first stayed it as the stream swept it down. A little lower, the current plunges, roaring, into a circular chasm like a well, surrounded on three sides by a chimney-like hollowness of polished rock, down which the foam slips in detached snow-flakes. Round the edges of the pool beneath, the water circles slowly, like black oil; a little butterfly lies on its back, its wings glued to one of the eddies, its limbs feebly quivering; a fish rises and it is gone. Lower down the stream, I can just see, over a knoll, the green and damp turf roofs of four or five hovels, built at the edge of a morass, which is trodden by the cattle into a black Slough of Despond at their doors, and traversed by a few ill-set stepping-stones, with here and there a flat slab on the tops, where they have sunk out of sight; and at the turn of the brook I see a man fishing, with a boy and a dog—a picturesque and pretty group enough certainly, if they had not been there all day
- 64. starving. I know them, and I know the dog’s ribs also, which are nearly as bare as the dead ewe’s; and the child’s wasted shoulders, cutting his old tartan jacket through, so sharp are they. We will go down and talk with the man. § 12. Or, that I may not piece pure truth with fancy, for I have none of his words set down, let us hear a word or two from another such, a Scotchman also, and as true hearted, and in just as fair a scene. I write out the passage, in which I have kept his few sentences, word for word, as it stands in my private diary:—“22nd April (1851). Yesterday I had a long walk up the Via Gellia, at Matlock, coming down upon it from the hills above, all sown with anemones and violets, and murmuring with sweet springs. Above all the mills in the valley, the brook, in its first purity, forms a small shallow pool, with a sandy bottom covered with cresses, and other water plants. A man was wading in it for cresses as I passed up the valley, and bade me good-day. I did not go much farther; he was there when I returned. I passed him again, about one hundred yards, when it struck me I might as well learn all I could about watercresses: so I turned back. I asked the man, among other questions, what he called the common weed, something like watercress, but with a serrated leaf, which grows at the edge of nearly all such pools. ‘We calls that brooklime, hereabouts,’ said a voice behind me. I turned, and saw three men, miners or manufacturers—two evidently Derbyshire men, and respectable-looking in their way; the third, thin, poor, old, and harder- featured, and utterly in rags. ‘Brooklime?’ I said. ‘What do you call it lime for?’ The man said he did not know, it was called that. ‘You’ll find that in the British ‘Erba,’ said the weak, calm voice of the old man. I turned to him in much surprise; but he went on saying something drily (I hardly understood what) to the cress-gatherer; who contradicting him, the old man said he ‘didn’t know fresh water,’ he ‘knew enough of sa’t.’ ‘Have you been a sailor?’ I asked. ‘I was a sailor for eleven years and ten months of my life,’ he said, in the same strangely quiet manner. ‘And what are you now?’ ‘I lived for ten years after my wife’s death by picking up rags and
- 65. bones; I hadn’t much occasion afore.’ ‘And now how do you live?’ ‘Why, I lives hard and honest, and haven’t got to live long,’ or something to that effect. He then went on, in a kind of maundering way, about his wife. ‘She had rheumatism and fever very bad; and her second rib grow’d over her hench-bone. A’ was a clever woman, but a’ grow’d to be a very little one’ (this with an expression of deep melancholy). ‘Eighteen years after her first lad she was in the family-way again, and they had doctors up from Lunnon about it. They wanted to rip her open and take the child out of her side. But I never would give my consent.’ (Then, after a pause:) ‘She died twenty-six hours and ten minutes after it. I never cared much what come of me since; but I know that I shall soon reach her; that’s a knowledge I would na gie for the king’s crown.’ ‘You are a Scotchman, are not you?’ I asked. ‘I’m from the Isle of Skye, sir; I’m a McGregor.’ I said something about his religious faith. ‘Ye’ll know I was bred in the Church of Scotland, sir,’ he said, ‘and I love it as I love my own soul; but I think thae Wesleyan Methodists ha’ got salvation among them, too.’” Truly, this Highland and English hill-scenery is fair enough; but has its shadows; and deeper coloring, here and there, than that of heath and rose. § 13. Now, as far as I have watched the main powers of human mind, they have risen first from the resolution to see fearlessly, pitifully, and to its very worst, what these deep colors mean, wheresoever they fall; not by any means to pass on the other side looking pleasantly up to the sky, but to stoop to the horror, and let the sky, for the present, take care of its own clouds. However this may be in moral matters, with which I have nothing here to do, in my own field of inquiry the fact is so; and all great and beautiful work has come of first gazing without shrinking into the darkness. If, having done so, the human spirit can, by its courage and faith, conquer the evil, it rises into conceptions of victorious and consummated beauty. It is then the spirit of the highest Greek and Venetian Art. If unable to conquer the evil, but remaining in strong,
- 66. though melancholy war with it, not rising into supreme beauty, it is the spirit of the best northern art, typically represented by that of Holbein and Durer. If, itself conquered by the evil, infected by the dragon breath of it, and at last brought into captivity, so as to take delight in evil for ever, it becomes the spirit of the dark, but still powerful sensualistic art, represented typically by that of Salvator. We must trace this fact briefly through Greek, Venetian, and Dureresque art; we shall then see how the art of decline came of avoiding the evil, and seeking pleasure only; and thus obtain, at last, some power of judging whether the tendency of our own contemplative art be right or ignoble. § 14. The ruling purpose of Greek poetry is the assertion of victory, by heroism, over fate, sin, and death. The terror of these great enemies is dwelt upon chiefly by the tragedians. The victory over them by Homer. The adversary chiefly contemplated by the tragedians is Fate, or predestinate misfortune. And that under three principal forms. a. Blindness, or ignorance; not in itself guilty, but inducing acts which otherwise would have been guilty; and leading, no less than guilt, to destruction.2 b. Visitation upon one person of the sin of another. c. Repression, by brutal or tyrannous strength, of a benevolent will. § 15. In all these cases sorrow is much more definitely connected with sin by the Greek tragedians than by Shakspere. The “fate” of Shakspere is, indeed, a form of blindness, but it issues in little more than haste or indiscretion. It is, in the literal sense, “fatal,” but hardly criminal. The “I am fortune’s fool” of Romeo, expresses Shakspere’s primary idea of tragic circumstance. Often his victims are entirely innocent, swept away by mere current of strong encompassing calamity (Ophelia, Cordelia, Arthur, Queen Katharine). This is rarely so with the Greeks. The victim may indeed be innocent, as Antigone, but is in some way
- 67. resolutely entangled with crime, and destroyed by it, as if it struck by pollution, no less than participation. The victory over sin and death is therefore also with the Greek tragedians more complete than with Shakspere. As the enemy has more direct moral personality,—as it is sinfulness more than mischance, it is met by a higher moral resolve, a greater preparation of heart, a more solemn patience and purposed self-sacrifice. At the close of a Shakspere tragedy nothing remains but dead march and clothes of burial. At the close of a Greek tragedy there are far-off sounds of a divine triumph, and a glory as of resurrection.3 § 16. The Homeric temper is wholly different. Far more tender, more practical, more cheerful; bent chiefly on present things and giving victory now, and here, rather than in hope, and hereafter. The enemies of mankind, in Homer’s conception, are more distinctly conquerable; they are ungoverned passions, especially anger, and unreasonable impulse generally (ἀτὴ). Hence the anger of Achilles, misdirected by pride, but rightly directed by friendship, is the subject of the Iliad. The anger of Ulysses (Ὀδυσσεὺς “the angry”), misdirected at first into idle and irregular hostilities, directed at last to execution of sternest justice, is the subject of the Odyssey. Though this is the central idea of the two poems, it is connected with general display of the evil of all unbridled passions, pride, sensuality, indolence, or curiosity. The pride of Atrides, the passion of Paris, the sluggishness of Elpenor, the curiosity of Ulysses himself about the Cyclops, the impatience of his sailors in untying the winds, and all other faults or follies, down to that—(evidently no small one in Homer’s mind)— of domestic disorderliness, are throughout shown in contrast with conditions of patient affection and household peace. Also, the wild powers and mysteries of Nature are in the Homeric mind among the enemies of man; so that all the labors of Ulysses are an
- 68. expression of the contest of manhood, not only with its own passions or with the folly of others, but with the merciless and mysterious powers of the natural world. § 17. This is perhaps the chief signification of the seven years’ stay with Calypso, “the concealer.” Not, as vulgarly thought, the concealer of Ulysses, but the great concealer—the hidden power of natural things. She is the daughter of Atlas and the Sea (Atlas, the sustainer of heaven, and the Sea, the disturber of the Earth). She dwells in the island of Ogygia (“the ancient or venerable”). (Whenever Athens, or any other Greek city, is spoken of with any peculiar reverence, it is called “Ogygian.”) Escaping from this goddess of secrets, and from other spirits, some of destructive natural force (Scylla), others signifying the enchantment of mere natural beauty (Circe, daughter of the Sun and Sea), he arrives at last at the Phæacian land, whose king is “strength with intellect,” and whose queen, “virtue.” These restore him to his country. § 18. Now observe that in their dealing with all these subjects the Greeks never shrink from horror; down to its uttermost depth, to its most appalling physical detail, they strive to sound the secrets of sorrow. For them there is no passing by on the other side, no turning away the eyes to vanity from pain. Literally, they have not “lifted up their souls unto vanity.” Whether there be consolation for them or not, neither apathy nor blindness shall be their saviours; if, for them, thus knowing the facts of the grief of earth, any hope, relief, or triumph may hereafter seem possible,—well; but if not, still hopeless, reliefless, eternal, the sorrow shall be met face to face. This Hector, so righteous, so merciful, so brave, has, nevertheless, to look upon his dearest brother in miserablest death. His own soul passes away in hopeless sobs through the throat-wound of the Grecian spear. That is one aspect of things in this world, a fair world truly, but having, among its other aspects, this one, highly ambiguous. § 19. Meeting it boldly as they may, gazing right into the skeleton face of it, the ambiguity remains; nay, in some sort gains upon them. We
- 69. trusted in the gods;—we thought that wisdom and courage would save us. Our wisdom and courage themselves deceive us to our death. Athena had the aspect of Deiphobus—terror of the enemy. She has not terrified him, but left us, in our mortal need. And, beyond that mortality, what hope have we? Nothing is clear to us on that horizon, nor comforting. Funeral honors; perhaps also rest; perhaps a shadowy life—artless, joyless, loveless. No devices in that darkness of the grave, nor daring, nor delight. Neither marrying nor giving in marriage, nor casting of spears, nor rolling of chariots, nor voice of fame. Lapped in pale Elysian mist, chilling the forgetful heart and feeble frame, shall we waste on forever? Can the dust of earth claim more of immortality than this? Or shall we have even so much as rest? May we, indeed, lie down again in the dust, or have our sins not hidden from us even the things that belong to that peace? May not chance and the whirl of passion govern us there; when there shall be no thought, nor work, nor wisdom, nor breathing of the soul?4 Be it so. With no better reward, no brighter hope, we will be men while we may: men, just, and strong, and fearless, and up to our power, perfect. Athena herself, our wisdom and our strength, may betray us;— Phœbus, our sun, smite us with plague, or hide his face from us helpless; —Jove and all the powers of fate oppress us, or give us up to destruction. While we live, we will hold fast our integrity; no weak tears shall blind us, no untimely tremors abate our strength of arm nor swiftness of limb. The gods have given us at least this glorious body and this righteous conscience; these will we keep bright and pure to the end. So may we fall to misery, but not to baseness; so may we sink to sleep, but not to shame. § 20. And herein was conquest. So defied, the betraying and accusing shadows shrank back; the mysterious horror subdued itself to majestic sorrow. Death was swallowed up in victory. Their blood, which seemed to be poured out upon the ground, rose into hyacinthine flowers. All the
- 70. beauty of earth opened to them; they had ploughed into its darkness, and they reaped its gold; the gods, in whom they had trusted through all semblance of oppression, came down to love them and be their helpmates. All nature round them became divine,—one harmony of power and peace. The sun hurt them not by day, nor the moon by night; the earth opened no more her jaws into the pit; the sea whitened no more against them the teeth of his devouring waves. Sun, and moon, and earth, and sea,—all melted into grace and love; the fatal arrows rang not now at the shoulders of Apollo the healer; lord of life and of the three great spirits of life—Care, Memory, and Melody. Great Artemis guarded their flocks by night; Selene kissed in love the eyes of those who slept. And from all came the help of heaven to body and soul; a strange spirit lifting the lovely limbs; strange light glowing on the golden hair; and strangest comfort filling the trustful heart, so that they could put off their armor, and lie down to sleep,—their work well done, whether at the gates of their temples5 or of their mountains;6 accepting the death they once thought terrible, as the gift of Him who knew and granted what was best. 1 Matt. v. 14. 2 The speech of Achilles to Priam expresses this idea of fatality and submission clearly, there being two vessels—one full of sorrow, the other of great and noble gifts (a sense of disgrace mixing with that of sorrow, and of honor with that of joy), from which Jupiter pours forth the destinies of men; the idea partly corresponding to the scriptural—“In the hand of the Lord there is a cup, and the wine is red; it is full mixed, and He poureth out of the same.” But the title of the gods, nevertheless, both with Homer and Hesiod, is given not from the cup of sorrow, but of good; “givers of good” (δωτὴρες ἐάων).— Hes. Theog. 664: Odyss. viii. 325. 3 The Alcestis is perhaps the central example of the idea of all Greek drama. 4 τῷ καὶ τεθνειῶτι νόον πόρε Περσεφόνεια, οἴω πεπνύσθαί τοὶ δὲ σκιαὶ ἀἴσσουσιν.
- 71. Od. x. 495. 5 οὐκέτι ὰνέστησαν, αλλ᾽ ἐν τέλει τουτῳ ἔσχοντο. Herod, i. 31. 6 ὁ δὲ ὰποπεμπόμενος, αὐτὸς μὲν οὐκ άπελίπετο τὸν δὲ παῖδα συστρατευόμενον, ἐόντα οἱ μουνογενέα, ἀπέπεμψε. Herod, vii. 221. CHAPTER III. THE WINGS OF THE LION. § 1. Such being the heroic spirit of Greek religion and art, we may now with ease trace the relations between it and that which animated the Italian, and chiefly the Venetian, schools. Observe, all the nobleness, as well as the faults, of the Greek art were dependent on its making the most of this present life. It might do so in the Anacreontic temper—Τί Πλειάδεσσι, κᾀμοί; “What have I to do with the Pleiads?” or in the defiant or the trustful endurance of fate;—but its dominion was in this world. Florentine art was essentially Christian, ascetic, expectant of a better world, and antagonistic, therefore, to the Greek temper. So that the Greek element, once forced upon it, destroyed it. There was absolute incompatibility between them. Florentine art, also, could not produce
- 72. landscape. It despised the rock, the tree, the vital air itself, aspiring to breathe empyreal air. Venetian art began with the same aim and under the same restrictions. Both are healthy in the youth of art. Heavenly aim and severe law for boyhood; earthly work and fair freedom for manhood. § 2. The Venetians began, I repeat, with asceticism; always, however, delighting in more massive and deep color than other religious painters. They are especially fond of saints who have been cardinals, because of their red hats, and they sunburn all their hermits into splendid russet brown. They differed from the Pisans in having no Maremma between them and the sea; from the Romans, in continually quarrelling with the Pope; and from the Florentines in having no gardens. They had another kind of garden, deep-furrowed, with blossom in white wreaths—fruitless. Perpetual May therein, and singing of wild, nestless birds. And they had no Maremma to separate them from this garden of theirs. The destiny of Pisa was changed, in all probability, by the ten miles of marsh-land and poisonous air between it and the beach. The Genoese energy was feverish; too much heat reflected from their torrid Apennine. But the Venetian had his free horizon, his salt breeze, and sandy Lido-shore; sloped far and flat,—ridged sometimes under the Tramontane winds with half a mile’s breadth of rollers;—sea and sand shrivelled up together in one yellow careering field of fall and roar. § 3. They were, also, we said, always quarrelling with the Pope. Their religious liberty came, like their bodily health, from that wave-training; for it is one notable effect of a life passed on shipboard to destroy weak beliefs in appointed forms of religion. A sailor may be grossly superstitious, but his superstitions will be connected with amulets and omens, not cast in systems. He must accustom himself, if he prays at all, to pray anywhere and anyhow. Candlesticks and incense not being
- 73. portable into the maintop, he perceives those decorations to be, on the whole, inessential to a maintop mass. Sails must be set and cables bent, be it never so strict a saint’s day, and it is found that no harm comes of it. Absolution on a lee-shore must be had of the breakers, it appears, if at all, and they give it plenary and brief, without listening to confession. Whereupon our religious opinions become vague, but our religious confidences strong; and the end of it all is that we perceive the Pope to be on the other side of the Apennines, and able, indeed, to sell indulgences, but not winds, for any money. Whereas, God and the sea are with us, and we must even trust them both, and take what they shall send. § 4. Then, farther. This ocean-work is wholly adverse to any morbid conditions of sentiment. Reverie, above all things, is forbidden by Scylla and Charybdis. By the dogs and the depths, no dreaming! The first thing required of us is presence of mind. Neither love, nor poetry, nor piety, must ever so take up our thoughts as to make us slow or unready. In sweet Val d’Arno it is permissible enough to dream among the orange- blossoms, and forget the day in twilight of ilex. But along the avenues of the Adrian waves there can be no careless walking. Vigilance, might and day, required of us, besides learning of many practical lessons in severe and humble dexterities. It is enough for the Florentine to know how to use his sword and to ride. We Venetians, also, must be able to use our swords, and on ground which is none of the steadiest; but, besides, we must be able to do nearly everything that hands can turn to—rudders, and yards, and cables, all needing workmanly handling and workmanly knowledge, from captain as well as from men. To drive a nail, lash a spear, reef a sail—rude work this for noble hands; but to be done sometimes, and done well, on pain of death. All which not only takes mean pride out of us, and puts nobler pride of power in its stead; but it tends partly to soothe, partly to chasten, partly to employ and direct, the hot Italian temper, and make us every way greater, calmer, and happier.
- 74. § 5. Moreover, it tends to induce in us great respect for the whole human body; for its limbs, as much as for its tongue or its wit. Policy and eloquence are well; and, indeed, we Venetians can be politic enough, and can speak melodiously when we choose; but to put the helm up at the right moment is the beginning of all cunning—and for that we need arm and eye;—not tongue. And with this respect for the body as such, comes also the sailor’s preference of massive beauty in bodily form. The landsmen, among their roses and orange-blossoms, and chequered shadows of twisted vine, may well please themselves with pale faces, and finely drawn eyebrows, and fantastic braiding of hair. But from the sweeping glory of the sea we learn to love another kind of beauty; broad- breasted; level-browed, like the horizon;—thighed and shouldered like the billows;—footed like their stealing foam;—bathed in cloud of golden hair, like their sunsets. § 6. Such were the physical influences constantly in operation on the Venetians; their painters, however, were partly prepared for their work by others in their infancy. Associations connected with early life among mountains softened and deepened the teaching of the sea; and the wildness of form of the Tyrolese Alps gave greater strength and grotesqueness to their imaginations than the Greek painters could have found among the cliffs of the Ægean. Thus far, however, the influences on both are nearly similar. The Greek sea was indeed less bleak, and the Greek hills less grand; but the difference was in degree rather than in the nature of their power. The moral influences at work on the two races were far more sharply opposed. § 7. Evil, as we saw, had been fronted by the Greek, and thrust out of his path. Once conquered, if he thought of it more, it was involuntarily, as we remember a painful dream, yet with a secret dread that the dream might return and continue for ever. But the teaching of the church in the middle ages had made the contemplation of evil one of the duties of men. As sin, it was to be duly thought upon, that it might be confessed.
- 75. As suffering, endured joyfully, in hope of future reward. Hence conditions of bodily distemper which an Athenian would have looked upon with the severest contempt and aversion, were in the Christian church regarded always with pity, and often with respect; while the partial practice of celibacy by the clergy, and by those over whom they had influence,— together with the whole system of conventual penance and pathetic ritual (with the vicious reactionary tendencies necessarily following), introduced calamitous conditions both of body and soul, which added largely to the pagan’s simple list of elements of evil, and introduced the most complicated states of mental suffering and decrepitude. § 8. Therefore the Christian painters differed from the Greek in two main points. They had been taught a faith which put an end to restless questioning and discouragement. All was at last to be well—and their best genius might be peacefully given to imagining the glories of heaven and the happiness of its redeemed. But on the other hand, though suffering was to cease in heaven, it was to be not only endured, but honored upon earth. And from the Crucifixion, down to a beggar’s lameness, all the tortures and maladies of men were to be made, at least in part, the subjects of art. The Venetian was, therefore, in his inner mind, less serious than the Greek: in his superficial temper, sadder. In his heart there was none of the deep horror which vexed the soul of Æschylus or Homer. His Pallas-shield was the shield of Faith, not the shield of the Gorgon. All was at last to issue happily; in sweetest harpings and seven-fold circles of light. But for the present he had to dwell with the maimed and the blind, and to revere Lazarus more than Achilles. § 9. This reference to a future world has a morbid influence on all their conclusions. For the earth and all its natural elements are despised. They are to pass away like a scroll. Man, the immortal, is alone revered; his work and presence are all that can be noble or desirable. Men, and fair architecture, temples and courts such as may be in a celestial city, or the clouds and angels of Paradise; these are what we must paint when we
- 76. want beautiful things. But the sea, the mountains, the forests, are all adverse to us,—a desolation. The ground that was cursed for our sake;— the sea that executed judgment on all our race, and rages against us still, though bridled;—storm-demons churning it into foam in nightly glare on Lido, and hissing from it against our palaces. Nature is but a terror, or a temptation. She is for hermits, martyrs, murderers,—for St. Jerome, and St. Mary of Egypt, and the Magdalen in the desert, and monk Peter, falling before the sword. § 10. But the worst point we have to note respecting the spirit of Venetian landscape is its pride. It was observed in the course of the third volume how the mediæval temper had rejected agricultural pursuits, and whatever pleasures could come of them. At Venice this negation had reached its extreme. Though the Florentines and Romans had no delight in farming, they had in gardening. The Venetian possessed, and cared for, neither fields nor pastures. Being delivered, to his loss, from all the wholesome labors of tillage, he was also shut out from the sweet wonders and charities of the earth, and from the pleasant natural history of the year. Birds and beasts, and times and seasons, all unknown to him. No swallow chattered at his window,1 nor, nested under his golden roofs, claimed the sacredness of his mercy;2 no Pythagorean fowl taught him the blessings of the poor,3 nor did the grave spirit of poverty rise at his side to set forth the delicate grace and honor of lowly life.4 No humble thoughts of grasshopper sire had he, like the Athenian; no gratitude for gifts of olive; no childish care for figs, any more than thistles. The rich Venetian feast had no need of the figtree spoon.5 Dramas about birds, and wasps, and frogs, would have passed unheeded by his proud fancy; carol or murmur of them had fallen unrecognized on ears accustomed only to grave syllables of war-tried men, and wash of songless wave.
- 77. § 11. No simple joy was possible to him. Only stateliness and power; high intercourse with kingly and beautiful humanity, proud thoughts, or splendid pleasures; throned sensualities, and ennobled appetites. But of innocent, childish, helpful, holy pleasures, he had none. As in the classical landscape, nearly all rural labor is banished from the Titianesque: there is one bold etching of a landscape, with grand ploughing in the foreground, but this is only a caprice; the customary Venetian background is without sign of laborious rural life. We find indeed often a shepherd with his flock, sometimes a woman spinning, but no division of fields, no growing crops nor nestling villages. In the numerous drawings and woodcuts variously connected with or representative of Venetian work, a watermill is a frequent object, a river constant, generally the sea. But the prevailing idea in all the great pictures I have seen, is that of mountainous land with wild but graceful forest, and rolling or horizontal clouds. The mountains are dark blue; the clouds glowing or soft gray, always massive; the light, deep, clear, melancholy; the foliage, neither intricate nor graceful, but compact and sweeping (with undulated trunks), dividing much into horizontal flakes, like the clouds; the ground rocky and broken somewhat monotonously, but richly green with wild herbage; here and there a flower, by preference white or blue, rarely yellow, still more rarely red. § 12. It was stated that this heroic landscape of theirs was peopled by spiritual beings of the highest order. And in this rested the dominion of the Venetians over all later schools. They were the last believing school of Italy. Although, as I said above, always quarrelling with the Pope, there is all the more evidence of an earnest faith in their religion. People who trusted the Madonna less, flattered the Pope more. But down to Tintoret’s time, the Roman Catholic religion was still real and sincere at Venice; and though faith in it was compatible with much which to us appears criminal or absurd, the religion itself was entirely sincere. § 13. Perhaps when you see one of Titian’s splendidly passionate subjects, or find Veronese making the marriage in Cana one blaze of
- 78. worldly pomp, you imagine that Titian must have been a sensualist, and Veronese an unbeliever. Put the idea from you at once, and be assured of this for ever;—it will guide you through many a labyrinth of life, as well as of painting,—that of an evil tree, men never gather good fruit—good of any sort or kind;— even good sensualism. Let us look to this calmly. We have seen what physical advantage the Venetian had, in his sea and sky; also what moral disadvantage he had, in scorn of the poor; now finally, let us see with what power he was invested, which men since his time have never recovered more. § 14. “Neither of a bramble bush, gather they grapes.” The great saying has twofold help for us. Be assured, first, that if it were bramble from which you gathered them, these are not grapes in your hand, though they look like grapes. Or if these are indeed grapes, it was no bramble you gathered them from, though it looked like one. It is difficult for persons, accustomed to receive, without questioning, the modern English idea of religion, to understand the temper of the Venetian Catholics. I do not enter into examination of our own feelings; but I have to note this one significant point of difference between us. § 15. An English gentleman, desiring his portrait, gives probably to the painter a choice of several actions, in any of which he is willing to be represented. As for instance, riding his best horse, shooting with his favorite pointer, manifesting himself in his robes of state on some great public occasion, meditating in his study, playing with his children, or visiting his tenants; in any of these or other such circumstances, he will give the artist free leave to paint him. But in one important action he would shrink even from the suggestion of being drawn. He will assuredly not let himself be painted praying.
- 79. Strangely, this is the action, which of all others, a Venetian desires to be painted in. If they want a noble and complete portrait, they nearly always choose to be painted on their knees. § 16. “Hypocrisy,” you say; and “that they might be seen of men.” If we examine ourselves, or any one else, who will give trustworthy answer on this point, so as to ascertain, to the best of our judgment, what the feeling is, which would make a modern English person dislike to be painted praying, we shall not find it, I believe, to be excess of sincerity. Whatever we find it to be, the opposite Venetian feeling is certainly not hypocrisy. It is often conventionalism, implying as little devotion in the person represented, as regular attendance at church does with us. But that it is not hypocrisy, you may ascertain by one simple consideration (supposing you not to have enough knowledge of the expression of sincere persons to judge by the portraits themselves). The Venetians, when they desired to deceive, were much too subtle to attempt it clumsily. If they assumed the mask of religion, the mask must have been of some use. The persons whom it deceived must, therefore, have been religious, and, being so, have believed in the Venetians’ sincerity. If therefore, among other contemporary nations with whom they had intercourse, we can find any, more religious than they, who were duped, or even influenced, by their external religiousness, we might have some ground for suspecting that religiousness to be assumed. But if we can find no one likely to have been deceived, we must believe the Venetian to have been, in reality, what there was no advantage in seeming. § 17. I leave the matter to your examination, forewarning you, confidently, that you will discover by severest evidence, that the Venetian religion was true. Not only true, but one of the main motives of their lives. In the field of investigation to which we are here limited, I will collect some of the evidence of this. For one profane picture by great Venetians, you will find ten of sacred subjects; and those, also, including their grandest, most labored, and
- 80. most beloved works. Tintoret’s power culminates in two great religious pictures: the Crucifixion, and the Paradise. Titian’s in the Assumption, the Peter Martyr, and Presentation of the Virgin. Veronese’s in the Marriage in Cana. John Bellini and Basaiti never, so far as I remember, painted any other than sacred subjects. By the Palmas, Vincenzo, Catena, and Bonifazio, I remember no profane subject of importance. § 18. There is, moreover, one distinction of the very highest import between the treatment of sacred subjects by Venetian painters and by all others. Throughout the rest of Italy, piety had become abstract, and opposed theoretically to worldly life; hence the Florentine and Umbrian painters generally separated their saints from living men. They delighted in imagining scenes of spiritual perfectness;—Paradises, and companies of the redeemed at the judgment;—glorified meetings of martyrs;— madonnas surrounded by circles of angels. If, which was rare, definite portraitures of living men were introduced, these real characters formed a kind of chorus or attendant company, taking no part in the action. At Venice all this was reversed, and so boldly as at first to shock, with its seeming irreverence, a spectator accustomed to the formalities and abstractions of the so-called sacred schools. The madonnas are no more seated apart on their thrones, the saints no more breathe celestial air. They are on our own plain ground—nay, here in our houses with us. All kind of worldly business going on in their presence, fearlessly; our own friends and respected acquaintances, with all their mortal faults, and in their mortal flesh, looking at them face to face unalarmed: nay, our dearest children playing with their pet dogs at Christ’s very feet. I once myself thought this irreverent. How foolishly! As if children whom He loved could play anywhere else. § 19. The picture most illustrative of this feeling is perhaps that at Dresden, of Veronese’s family, painted by himself.
- 81. He wishes to represent them as happy and honored. The best happiness and highest honor he can imagine for them is that they should be presented to the Madonna, to whom, therefore, they are being brought by the three virtues—Faith, Hope, and Charity. The Virgin stands in a recess behind two marble shafts, such as may be seen in any house belonging to an old family in Venice. She places the boy Christ on the edge of a balustrade before her. At her side are St. John the Baptist, and St. Jerome. This group occupies the left side of the picture. The pillars, seen sideways, divide it from the group formed by the Virtues, with the wife and children of Veronese. He himself stands a little behind, his hands clasped in prayer. § 20. His wife kneels full in front, a strong Venetian woman, well advanced in years. She has brought up her children in fear of God, and is not afraid to meet the Virgin’s eyes. She gazes steadfastly on them; her proud head and gentle, self-possessed face are relieved in one broad mass of shadow against a space of light, formed by the white robes of Faith, who stands beside her,—guardian, and companion. Perhaps a somewhat disappointing Faith at the first sight, for her face is not in any way exalted or refined. Veronese knew that Faith had to companion simple and slow-hearted people perhaps oftener than able or refined people—does not therefore insist on her being severely intellectual, or looking as if she were always in the best company. So she is only distinguished by her pure white (not bright white) dress, her delicate hand, her golden hair drifted in light ripples across her breast, from which the white robes fall nearly in the shape of a shield—the shield of Faith. A little behind her stands Hope; she also, at first, not to most people a recognizable Hope. We usually paint Hope as young, and joyous. Veronese knows better. That young hope is vain hope—passing away in rain of tears; but the Hope of Veronese is aged, assured, remaining when all else had been taken away. “For tribulation worketh patience, and
- 82. patience experience, and experience hope;” and that hope maketh not ashamed. She has a black veil on her head. Then again, in the front, is Charity, red-robed; stout in the arms,—a servant of all work, she; but small-headed, not being specially given to thinking; soft-eyed, her hair braided brightly, her lips rich red, sweet- blossoming. She has got some work to do even now, for a nephew of Veronese’s is doubtful about coming forward, and looks very humbly and penitently towards the Virgin—his life perhaps not having been quite so exemplary as might at present be wished. Faith reaches her small white hand lightly back to him, lays the tips of her fingers on his; but Charity takes firm hold of him by the wrist from behind, and will push him on presently, if he still hangs back. § 21. In front of the mother kneel her two eldest children, a girl of about sixteen, and a boy a year or two younger. They are both wrapt in adoration—the boy’s being the deepest. Nearer us, at their left side, is a younger boy, about nine years old—a black-eyed fellow, full of life—and evidently his father’s darling (for Veronese has put him full in light in the front; and given him a beautiful white silken jacket, barred with black, that nobody may ever miss seeing him to the end of time). He is a little shy about being presented to the Madonna, and for the present has got behind the pillar, blushing, but opening his black eyes wide; he is just summoning courage to peep round, and see if she looks kind. A still younger child, about six years old, is really frightened, and has run back to his mother, catching hold of her dress at the waist. She throws her right arm round him and over him, with exquisite instinctive action, not moving her eyes from the Madonna’s face. Last of all, the youngest child, perhaps about three years old, is neither frightened nor interested, but finds the ceremony tedious, and is trying to coax the dog to play with him; but the dog, which is one of the little curly, short-nosed, fringy- pawed things, which all Venetian ladies petted, will not now be coaxed.
- 83. For the dog is the last link in the chain of lowering feeling, and takes his doggish views of the matter. He cannot understand, first, how the Madonna got into the house; nor, secondly, why she is allowed to stay, disturbing the family, and taking all their attention from his dogship. And he is walking away, much offended. § 22. The dog is thus constantly introduced by the Venetians in order to give the fullest contrast to the highest tones of human thought and feeling. I shall examine this point presently farther, in speaking of pastoral landscape and animal painting; but at present we will merely compare the use of the same mode of expression in Veronese’s Presentation of the Queen of Sheba. § 23. This picture is at Turin, and is of quite inestimable value. It is hung high; and the really principal figure—the Solomon, being in the shade, can hardly be seen, but is painted with Veronese’s utmost tenderness, in the bloom of perfect youth, his hair golden, short, crisply curled. He is seated high on his lion throne; two elders on each side beneath him, the whole group forming a tower of solemn shade. I have alluded, elsewhere, to the principle on which all the best composers act, of supporting these lofty groups by some vigorous mass of foundation. This column of noble shade is curiously sustained. A falconer leans forward from the left-hand side, bearing on his wrist a snow-white falcon, its wings spread, and brilliantly relieved against the purple robe of one of the elders. It touches with its wings one of the golden lions of the throne, on which the light also flashes strongly; thus forming, together with it, the lion and eagle symbol, which is the type of Christ throughout mediæval work. In order to show the meaning of this symbol, and that Solomon is typically invested with the Christian royalty, one of the elders, by a bold anachronism, holds a jewel in his hand of the shape of a cross, with which he (by accident of gesture) points to Solomon; his other hand is laid on an open book.
- 84. § 24. The group opposite, of which the queen forms the centre, is also painted with Veronese’s highest skill; but contains no point of interest bearing on our present subject, except its connection by a chain of descending emotion. The Queen is wholly oppressed and subdued; kneeling, and nearly fainting, she looks up to Solomon with tears in her eyes; he, startled by fear for her, stoops forward from the throne, opening his right hand, as if to support her, so as almost to drop the sceptre. At her side her first maid of honor is kneeling also, but does not care about Solomon; and is gathering up her dress that it may not be crushed; and looking back to encourage a negro girl, who, carrying two toy-birds, made of enamel and jewels, for presenting to the King, is frightened at seeing her Queen fainting, and does not know what she ought to do; while lastly, the Queen’s dog, another of the little fringy- paws, is wholly unabashed by Solomon’s presence, or anybody else’s; and stands with his fore legs well apart, right in front of his mistress, thinking everybody has lost their wits; and barking violently at one of the attendants, who has set down a golden vase disrespectfully near him. § 25. Throughout these designs I want the reader to notice the purpose of representing things as they were likely to have occurred, down to trivial, or even ludicrous detail—the nobleness of all that was intended to be noble being so great that nothing could detract from it. A farther instance, however, and a prettier one, of this familiar realization, occurs in a Holy Family, by Veronese, at Brussels. The Madonna has laid the infant Christ on a projecting base of pillar, and stands behind, looking down on him. St. Catherine, having knelt down in front, the child turns round to receive her—so suddenly, and so far, that any other child must have fallen over the edge of the stone. St. Catherine, terrified, thinking he is really going to fall, stretches out her arms to catch him. But the Madonna looking down, only smiles, “He will not fall.” § 26. A more touching instance of this realization occurs, however, in the treatment of the saint Veronica (in the Ascent to Calvary), at
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