On Edge Control Set of a Graph in Transportation Problems

Int. J. Advanced Networking and Applications
Volume: 08 Issue: 01 Pages: 3003-3008 (2016) ISSN: 0975-0290 3003
On Edge Control Set of a Graph in Transportation
Problems
Manoshi Kotoky
Department of Mathematics, Dibrugarh University, Dibrugarh : 786004, Assam, India
E-mail : mkotoky@yahoo.com
Arun Kumar Baruah
Department of Mathematics, Dibrugarh University, Dibrugarh : 786004, Assam, India
E-mail : baruah_arun123@rediffmail.com
-------------------------------------------------------------------ABSTRACT--------------------------------------------------------------
One of the most significant problems in the analysis of the reliability of multi-state transportation systems is to find the
minimal cut sets and minimal edge control sets. For that purpose there are several algorithms that use the minimal path
and cut sets of such systems. In this paper we give an approach to determine the minimal edge control set. This approach
directly finds all minimal edge control sets of a transport network. The main aim of the paper is to find optimal locations
for sensors for detecting terrorists, weapons, or other dangerous materials on roads leading into major cities.
Keywords: Edge Control Set, Minimal Edge Control Set, Sensors, Transport network.
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Date of submission: Aug 03, 2016 Date of Acceptance: Aug 18, 2016
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1. INTRODUCTION
One of the most important and successful applications
of quantitative analysis in solving business problems
has been in the physical distribution of products,
commonly referred to as transportation problems.
Basically, the purpose is to minimize the cost of
shipping goods from one location to another so that the
needs of each arrival area are met and every shipping
location operates within its capacity [1] [2].
Transportation problem is one of the subclasses of
LPP’s in which the objective is to transport various
quantities of a single homogeneous commodity that are
initially stored at various sources to different
destinations in a way to minimize the total
transportation cost, time, distance etc. These types of
problems can be solved by general network methods.
Transportation problems belong to a special class of
network flow problems. Although these problems can
be formulated as linear programming models, it is much
more natural to formulate them in terms of nodes and
arcs, taking advantage of the special structure of the
problem. However, quantitative analysis has been used
for many problems other than the physical distribution
of goods. For example, it has been used to efficiently
place employees at certain jobs within an organization,
called an assignment problem [2].
Networks are essential components of our national
infrastructure. Those networks could be used by
terrorists seeking to attack dense urban populations with
weapons of mass destruction. In particular, large urban
road networks provide many routes that terrorists could
use to get close enough to a major city to make a
harmful attack. One approach envisioned for protecting
urban areas from such attack is to deploy human-
operated or fully automatic sensors on the roads, around
cities to detect terrorists and their weapons so that they
can be stopped before they come within range of their
targets [3]. A key challenge to such an approach
concerns how many sensors are to buy and where to
locate them. Indeed, the size and density of road
networks would seem to make the cost of buying and
operating these sensors, prohibitive by requiring
placement of sensors on hundreds if not thousands of
road segments in order to protect any large city [2].
This challenge led to the work reported here, which
shows that, the number of sensors required to cover
every possible route into a city is not prohibitively
large. We apply graph theory to find a minimal edge
control set for a road network; i.e., to find a smallest set
of road segments on which sensors must be placed to
ensure that a terrorist traveling across the road network
must encounter at least one sensor [4] [5] [6]. There are
two situations occur when we use minimal edge control
set to a connected network. For some case if we remove
the minimal edge control set from the network, the
remaining graph will be disconnected and for some
cases it is connected. In this paper both the cases are
discussed.
The work reported here specifically concerns finding
optimal locations for sensors for detecting terrorists,
weapons, or other dangerous materials on roads leading
into major cities. However, this work is generally
applicable to finding minimal edge control sets for any
large network. It could be used to find optimal sensor
locations on other transportation networks like railroads
or subways. It could also be used to support offensive
operations by locating a smallest set of segments in an
adversary’s network that would have to be cut in order
to completely stop the flow through the network. Thus,
the methodology presented here could have utility in
other homeland security and military analysis.

2. Edge Control Set
To study the transportation problem, it has to be
modeled mathematically by using a simple graph. The
set of edges of the underlying graph will represent the
communication link between the set of nodes. In the
graph representing the transportation problem, the
vertices will be joined by an edge if there is a
communication link between the vertices otherwise not.
In order to define an edge control set of a graph, we
consider the underlying graph G = (V, E) where
V (G) denotes the set of vertices of G and E (G) denotes
the set of edges of G. A cut-set F of G is called an edge
control set of G if every flow of G is completely
determined by F.A subset F ⊆ E(G) is a cut-set of G if
the removal of F from G disconnects G [1], [3]. Also it
results in the increase in the number of components of
G by one.
3. Minimal Edge Control Set
Let G = (V, E) be a graph and E(G) denotes the set of
edges of G. An edge control set F is said to be minimal
if any proper subset of F is not an edge control set of
the graph G. As the edge control set of a graph is not
unique, therefore it is important to find the set with the
minimum number of edges.
Definition 3.1 Let G = (V, E) be a graph, let H be a sub
graph of G and e ЄE(H).We define
CH (e)= {e}∪ {d ЄE(H) : d is a cut edge of H–{e}}
Then CH ( e ) is called the control of e in H.
Algorithm 3.2
Let G be a graph and a subset F ⊆E(G) is constructed
by the following steps.
Step 1: Let F: =Ø and H: = G
Step 2: While E(H)≠Ø , select any edge e Є E(H)
F: =F∪ {�}, H: = H - CH (e).
Then F is the minimal edge control set of the graph G
[7] [8].
Proof of the Algorithm
Let G be a graph. Then to prove that the set F
constructed using the algorithm is the minimal edge
control set. Let F = { e1, e2, e3, ….et } be the edges
which are introduced to the set F in the same order as
they are labeled and
E(G) = E(H)⊒ E(H1 )⊒ E( H2). . . ⊒ E(Ht ) = Ø
be the sequence of sub graphs as they are generated
using the algorithm.
As the removal of the set F disconnects the graph,
therefore F is an edge control set of G and we are to
show that F is the minimal edge control set of the graph
G.
Let us suppose that there exist a set ⊆ , which is
also an edge control set of G. Since /
⊆ ,∃ an edge et
Є F which is not in /
. It implies that �� ,
which is the smallest sub graph of the sequence of sub
graph generated using the algorithm.
Since��
/
, then there exists an edge �� of the sub
graph � which is connected to somevertices of the
graph G and the removal of the set will not disconnect
the graph G. Hence,
E(G) = E(H) ⊃E(H1 )⊃ E(H2). . .⊃ E(Ht) ≠Ø
This implies that there exists at least one edge of G
which is connected to some vertices of the graph G.
Therefore /
cannot be an edge control set of G which
is a contradiction to` /
⊆ . Hence F is a minimal edge
control set of G constructed by the algorithm.
4. EXAMPLES
(i) Disconnected Case:
Let us consider a transportation problem with road
segments as shown in Fig. 1. Here nodes represent the
different places of a city and edges represent the roads
joining them. The corresponding graphs are shown in
the figures below:
To start with the Algorithm (3.2), we consider F: =
Ø and H: = G. Now we select any edge e1 such that
e1ЄE (H) i.e. E(H) ≠Ø. Thus we have,
CH(e1) = { e1 } U { e2 }
Fig. 1: A graph with thirteen road segments

= {e1, e2 }
F =F U { e1} ={ e1 }
H: =H-CH (e1)
= { e3, e4, e5, e6, e7, e8, e9, e10, e11, e12, e13 }
Where the first subgraph H: is as shown in Fig. 2
below:
Again since E (H) ≠Ø, let us select any edge e5ЄE
(H). Then we obtain
CH ( e5)= { e5 } U { e4 }
= { e4, e5 }
F ={ e1 } U { e5 } ={ e1, e5 }
H: =H – CH ( e5 )
= { e3, e6, e7, e8, e9, e10, e11, e12, e13 }
Where the second subgraph H: is as shown in Fig. 3
below:
Since E (H) ≠Ø, let us select any edge e10ЄE (H). Then
we obtain
CH ( e10 )= { e10 } U { e8, e9, e11, e12 }
= { e8, e9, e10, e11, e12 }
F = { e1, e5 } U { e10 } = { e1, e5 , e10 }
H: =H – CH ( e10 )
= { e3, e6, e7, e13 }
Where the third subgraph H: is as shown in Fig. 4
below:
Now, let e6 Є E (H), then we obtain
CH ( e6 )= { e6 } U { e7 }
= { e6, e7 }
F = { e1,e5 ,e10 } U { e6 } ={ e1, e5,e6,e10 }
H: =H – CH( e6 )
= { e3, e13 }
Where the fourth subgraph H: is as shown in Fig. 5
below:
Fig. 2 : H is the first subgraph obtained
applying the Algorithm (3.2)
Fig. 3 : H is the second subgraph obtained
Fig. 4: H is the third subgraph obtained applying
the Algorithm (3.2)
Fig. 5 : H is the fourth subgraph obtained

Finally, let e3ЄE(H), then we obtain CH
(e3)={ e3, e13 } and F={ e1,e3 e5, e6, e10 }. The new H:
consists of all isolated vertices, i.e., E (H) =Ø. The fifth
subgraph with all isolated vertices is shown in Fig. 6
Therefore F= { e1, e3, e5, e6, e10 } is a minimal edge
control set of G. The graph obtained after removing F
from G is as shown in Fig.7 below:
Finally, it is also clear from the above diagram that the
above graph is disconnected having two components
and no other set having less number of edges than F
disconnect the graph, therefore F is a minimal edge
control set of the graph G constructed using the
algorithm. From the above discussion it is clear that
sensors have to be placed in the edges e1, e3, e5, e6 and
e10 which will provide complete information about the
whole transportation network.
(ii) Connected Case:
Let us consider a transportation problem with road
segments as shown in Fig. 8. Here nodes represent the
different places of a city and edges represent the roads
joining them. The corresponding graphs are shown in
figures below:
To start with the Algorithm (3.2), we consider F: =
Ø and H: = G. Now we select any edge e1 such that
e1ЄE (H) i.e. E(H) ≠Ø. Thus we have,
CH (e1) = { e1 } U { e6, e7 }
= {e1, e6, e8 }
F = F U { e1 } ={ e1 }
H: = H-CH ( e1 )
= { e2, e3, e4, e5, e6, e8, e9, e10, e11, e12 }
Where the first subgraph H: is as shown in Fig. 9
below:
Fig. 6 : H is the fifth and final subgraph
obtained applying the Algorithm (3.2)
Fig. 8: A graph with twelve road segments
Fig. 7: Graph obtained after removing minimal
edge control set

Let e2Є E (H), then we obtain
CH ( e2 )= { e2 } U { e9 }
= { e2, e9 }
F ={ e1 } U { e2 } ={ e1, e2 }
H: =H – CH ( e2 )
= { e3, e4, e, e8, e10, e11, e12 }
Where the second subgraph H: is as shown in Fig. 10
below:
Let e3 Є E(H), then we obtain
CH ( e3)= { e3 } U { e10 }
= { e3, e10 }
F = { e1, e2 } U { e3 } ={ e1, e2, e3 }
H: =H – CH ( e3 )
= { e4,e5, e8, e11, e12 }
Where the third subgraph H: is as shown in Fig. 11
below:
Let e4Є E(H), then we obtain
CH ( e4)= { e4} U { e8}
= { e4, e8}
F = { e1, e2, e3} U { e4 } ={ e1, e2, e3, e4 }
H: =H – CH( e4 )
= { e5, e11, e12}
Where the fourth subgraph H: is as shown in Fig.12
below:
Let e5Є E(H), then we obtain
CH ( e5)= { e5} U { e11, e12}
= { e5, e11, e12}
F = { e1, e2, e3 } U { e5 } ={ e1, e2, e3, e4, e5 }
H: =Ø
Fig. 9 : H is the first subgraph obtained
Fig. 12 : H is the fourth subgraph obtained
Fig. 10: H is the second subgraph obtained
Fig. 11 : H is the third subgraph obtained

The third subgraph with all isolated vertices is shown in
Fig. 13
Therefore F= { e1,e2 e3, e4, e5 } is a minimal edge control
set of G. The graph obtained after removing F from G
is as shown in Fig. 14 below:
Finally, it is also clear from the above diagram that F is
a minimal edge control set of the graph G constructed
using the algorithm. The graph remains connected even
after removal of F. From the above discussion it is clear
that sensors have to be placed in the edges e1, e2, e3, e4
and e5 which will provide complete information about
the whole transportation network.
5. CONCLUSION
In this paper we have used minimal edge control set as
a graph theoretic tool to study the transportation
problem. Minimal edge control set determines the
whole transportation flow. This can be achieved by
placing traffic sensors on each of the minimal edge
control set of the transportation network which will
provide complete information of the transport network.
Here two examples are considered to explain the use of
minimal edge control set. In one case minimal edge
control set is used as a cut set whose removal
disconnects the graph. If terrorist wanted to attack a
major city, they can be stopped by removing these
edges. In the other example we consider another
minimal edge control set whose removal does not
disconnect the network. The sensors can be placed on
each edge control set and from the definition of an edge
control set these sensors will provide complete
information for the control system. Thus optional
locations for the traffic sensors can be obtained by
using edge control set.
6. Acknowledgements
The authors would like to thank Professor Arun Kumar
Baruah for helpful suggestions which led to a major
revision of the paper.
REFERENCES
[1] D. B. West, Introduction to Graph Theory (2nd
Edition Prentice Hall Inc., New Jersey, 2001).
[2] R.K. Ahuja , T.L. Magnanti, and J.B. Orlin.
“Maximum Flows: Basic Ideas.” Chapter 6 of Network
Flows Theory, Algorithms, and Applications.Prentice
Hall, 1993.
[3] R. J. Wilson, J. J. Watkins, Graph : An Introductory
Approach (John Wiley & Sons, New York, 1989).
[4] V.K. Balakrishnan “Graph Theory.” Appendix of
CombinatoricsIncluding Concepts of Graph Theory,
Schaum’s Outline Series.McGraw-Hill, 1995.
[5] G.H. Bradley , G.G. Brown, and G.W. Graves.
“Design and Implementation of Large-Scale Primal
Transshipment Algorithms.” Management Science 24,
no. 1 (1977): 1-34.
[6] M.O. Ball “Design of Survivable Networks.”
Chapter 10 of Handbooks in OR & MS, Vol. 7, Network
Models, M.O. Ball et al., Eds. Elsevier Science B.V.,
1995.
[7] W. Gu, X. Jia, On a Traffic Sensing Problem,
Preprint, Texas State University, 2006.
[8] W. Gu, X. Jia, On a Traffic Control Problem, 8th
International Symposium on Parallel Architectures,
Algorithms and Networks (ISPAN’ 05), 2005.
Fig. 13 : H is the fifth and final subgraph obtained
Fig. 14: Graph obtained after removing
minimal edge control set

On Edge Control Set of a Graph in Transportation Problems

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