Graphs data Structure

using Java
2015
Data Structure
Prepared by: Mahmoud Rafeek Al-farra
in Java
10. Graphs

mfarra.cst.ps www.fb.com/MahmoudRFarra
Contents
Representing Graphs
Basic Graph Terminologies
Introduction
Modeling Graphs (Implementation)

Introduction
 A graph is a nonlinear data structure with many
important applications.
 Many real-world problems can be solved using graph
algorithms.
 For example:
 The problem to find the least number of flights
between two cities can be modeled using a graph,
where the vertices represent cities and the edges
represent the flights between two adjacent
cities.

Introduction

Introduction
 The study of graph problems is known as graph
theory.
 There were two islands on the river. The city and
islands were connected by seven bridges.
 The question is, can one take a walk, cross each
bridge exactly once, and return to the starting
point?
 Euler proved that it is not possible.

Introduction
 Euler proved that it is not possible.

 A graph is a mathematical structure that represents
relationships among entities in the real world.
 A graph consists of a nonempty set of vertices (also
known as nodes or points), and a set of edges that
connect the vertices.
 For convenience:
 We define a graph as G = (V, E). Where
 V represents a set of vertices
 E represents a set of edges.

 For example, V and E for the graph in first figure
are as follows:

 A graph may be directed or undirected.
 In a directed graph, each edge has a direction, which
indicates that you can move from one vertex to the
other through the edge.
 You can model parent/child relationships using a
directed graph, where an edge from vertex A to B
indicates that A is a parent of B.

 A graph may has many forms.

Before we introduce graph algorithms and applications, it is helpful to
get acquainted with graphs using the interactive tool at
www.cs.armstrong.edu/liang/animation/GraphLearningTool.html
 In an undirected graph, you can move in both
directions between vertices.
 Edges may be weighted or unweighted.
 For example, you can assign a weight for each edge in
the graph. (e.x.: To indicate the flight time between
the two cities.)

 Two vertices in a graph are said to be adjacent:
 if they are connected by the same edge.
 (Edges) if they are connected to the same vertex.
 An edge in a graph that joins two vertices is said to
be incident to both vertices.
 The degree of a vertex is the number of edges
incident to it.

 Two vertices are called neighbors if they are
adjacent. Similarly, two edges are called neighbors if
they are adjacent.
 A loop is an edge that links a vertex to itself.
 If two vertices are connected by two or more edges,
these edges are called parallel edges.
 A simple graph is one that has
doesn’t have any loops or parallel
edges.
 In a complete graph, every two
pairs of vertices are connected, as
shown.

 A graph is connected if there exists a path between
any two vertices in the graph.
 A subgraph of a graph G is a graph whose vertex set
is a subset of that of G and whose edge set is a
subset of that of G.

Representing Graphs
 Representing a graph is to store its vertices and
edges in a program.
 The data structure for storing a graph is arrays or
lists.

Representing Graphs
 The edges can be represented using a two-
dimensional array.
 The vertices and edges in the following figure can be
represented as follows:

Modeling Graphs (Implementation)
 The Graph interface defines the common operations
for a graph.

using Java
2015
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Graphs data Structure

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Graphs data Structure