Literature one of the immensely researched branches is domination

Literature
Review

 

Any real-world
situation can be interpreted graphically
with a group of points joined with lines.
For example, a computer network can be represented graphically in which the
points represent computer terminals and lines representing communication links
between computer. A mathematical abstraction of real-world
situations which focuses on representing situation by using points,
which are connected together give rise to the concept called graph 36. Due
to a number of applications in computer and communication, biological sciences,
computational linguistics, molecular physics and chemistry, social networks,
and in other numerous areas, graph theory has become one of the most researched
areas of modern mathematics which has seen an exponential growth. In graph
theory, one of the immensely researched branches is domination in the graph 7.

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In the last three
decades, an exponential growth has been seen in graph theory because of its
wide range of applications in optimization problems, combinatorial problems,
classical algebraic problems, computational problems etc. This growth is mainly
due to the rise of a number of new parameters that have been developed from the basic definition of domination.

 

The intense research of
dominating set in graph theory has been started around the early 60’s. This
branch of graph theory has its historical roots during 1862 when C.F. De
Jaenisch started studying the problem of chessboard game moves in which he
tried determining the minimum number of queens that are needed to dominate an n × n chessboard. 

 

Soon in coming years,
W.W. Rouse Ball also reported three basic chess problem namely; covering
problem, independent covering problem and independence problems related to set.
W.W. Rouse Ball has started studying this  three problem in 1892. Further, in 1958, Claude Berge introduced the term domination
number of a graph.

 

In the year 1962, Ore has used the terms
dominating set and domination number for the same concept in graph theory 8
11.  The graph theorists E.J.Cockayne,
S.T. Hedetniemi, R.C. Laskar have made some of the interesting and broad
reviews on the results of dominating sets in graphs during 70’s. 

 

Other
graph theory researchers H.L. Abbott, Arumugam, B.D.Acharya, R.B. Allan,
D.W.Bange, A.E.Barkausks,  P.J. Slater,
C.Berge, A.A.Bertossi, B.Bollobas, D.G.Corneil, Y.Perl, G.S.Domke, M.Farber,
J.F. Fink, M.S. Jacobson, y.Fu, D.L.Grinstead, D.S. Hochbaum, T. Kikuno, Krishnamoorthy.,
M.S.,K.Murthy, A.Meir, J.W.Moon, H.Muller, C.M. Mynhardt, C.Payen, E.
Sampathkumar, H.B.Walikar, T.V.Wimer, B.Zelinka and many others 8 has done a
significant research in dominating set, domination numbers 13 15 and other
related topics in graph theory.

 

The
applications of dominating set and domination number of   the graph in computer networking  will be discussed in detail in the following
section.

 

4.2 Domination   Sets in Computer
Networks

 

Domination
is an area in graph theory with an extensive research activity. The dominating
set problem tells to determine the domination number of a graph. The concept of
dominating set occurs in a variety of problems. A number of problems are
motivated by communication network problems. The communication network includes
a set of nodes, where one node can communicate with another, if it is directly
connected to that node. In order to send a message directly from a set of nodes
to all others, one has to choose the set in such a way that all other nodes are
connected to at least one node in the set. Now, such a set is a dominating set
in a graph, which represents the network .Other applications of domination are
the facility location problem, land surveying and routings. Its application
lies in various fields of solving mathematical abstraction of real-life
problems

 

Social Network Theory

 

A social networking service is an
online platform that people use to build social networks or social relations with other people who
share similar personal or career interests, activities, backgrounds or
real-life connections.

 

The
online social network has been developed significantly and immensely grown in
the recent years as a medium of communication, sharing the information and
spreading the influence 1. The data from the social network can be used to
carry out research on the various social problem
and can be utilized in disseminating the information and ideas. The online social network can be utilized for
solving general physiological and social problems in the physical world such as
drinking, smoking, and drug problems are all explored well. 

 

The
dominating set plays a vital role in analyzing the effect on a real online
social network data set through
simulation.

 

The dominating set concept can be
applied to the social network by stimulating it as a graph in which point
represent user and links are used to represent relationship, this network graph
can be used to determine the amount of positive or negative influence that is
possessed by an individual as well as its impact on their related neighbor. This degree threshold positivity spread
by a dominating positive user can spread the positive educational influence
throughout the entire community in the social network.

 

The positive, as well as negative impact
of different people in a social setup, can play different roles as they are
affected by their peers. The positive or negative influence in social issues
can move in two directions. That is, a positive individual can convert into a
negative individual and can move back and forth between these two states for
multiple times, that is they can change their state by the influence of their
neighbor.

 

An undirected graph G = (V, E, C) is
used to denote the online social network, because friendship in an online
social network is usually bidirectional as shown in Figure 4.21.

Fig 4.21 Social Network Architecture Instance

 

Description
of the above figure: Online social network can be represented as a graph of
relationship with individuals representing the nodes of a graph (V), the social
interactions as edges (E) and C is the compartment vector that saves the
compartment of each node. The compartment part of a node decides whether the
social issue(s) of an individual has a positive
or negative impact on its neighbors.

 

Computer Communication Networks

 

The
dominating set plays a vital role in networking and information exchange in a
computer and communication networks to route the correct information between
the receiver and sending nodes. For example, wireless network. A wireless
network is a type of computer network that uses wireless data transmission by
connecting the wireless network nodes. The examples of a wireless network are cell-phone network, mobile
Ad-hoc network, wireless sensor network etc.

 

Dominating Set in Mobile Ad-hoc Network (MANET)

 

A
Mobile Ad-Hoc Network (MANET) is a self-configurable infrastructure-less network which is used for connecting the mobile
devices with accurate transmission link in wireless mode. Routing and
broadcasting the information between mobile devices in mobile ad-hoc networks
is studied and implemented using dominating set 18.  The Connected Dominating Set (CDS) is widely
used as a virtual backbone for MANETs to provide different communication
primitive such as routing, broadcasting etc 14. The construction of virtual
backbone is very important and it is approximated by a minimum CDS with a unit
disk graph 4 as shown in Figure 4.22

Fig. 4.22 CDS Construction in MANETs

 

The
connected dominating set is defined as a subset of nodes in a wireless network
such that each node in the network is either present in the set or as a neighbor of some node in the set and the
induced graph with the nodes in the set is connected. The CDS is useful in
reducing the communication and the storage overhead by keeping its size to be
minimal. However, it is well–known that the problem of finding the minimal
CDS is     NP–hard 19 and it can be solved practically through preprocessing
2.

Dominating Set in Wireless Sensor Network 

 

A Wireless Sensor Network (WSN) is a
type of wireless network which consists of spatially distributed autonomous
sensors to monitor the physical or environmental conditions such as
temperature, pressure, sound etc. and to broadcast their information through
the wireless network to a central location 19 20. 

 

The main activity in the WSN thus forms
the timely as well as correct routing of information between the nodes. Routing
in WSN is a very challenging because of the inherent characteristics of
distinguishing these networks from other wireless networks such as MANETs and
other cellular networks. A number of routing techniques are available for
wireless sensor networks. Among them, hierarchical routing and cluster-based
routing methods are well-known techniques
for scalable and efficient communication. The hierarchical routing with virtual
backbone infrastructure is mainly employed to perform energy–efficient routing
in WSN(s). Thus, the CDS comes into the picture
to serve as a virtual backbone for WSNs to reduce the routing overheads. The
CDS incorporated hierarchical techniques simplifies the routing by restricting
the main routing task to the dominating nodes only.

 

Minimal Communication   power consumption in Wireless Sensor Network

 

A
group of wireless sensors forms a wireless sensor network to become a very
responsive and broadly operating network. In a wireless network along with its
responsiveness and time consumption amount of power,
use plays a vital role in its credibility. A virtual backbone routing
environment of a wireless network consists of a connected
subset of nodes

which
is responsible for routing the information in the network. A node in that
subset is likely to get exhausted because of performing heavy duties as
compared to other nodes. This situation can get more intensified if the node
uses high communication power to form the virtual backbone. Hence to compute a
virtual backbone in a wireless network with minimal total communication power
the connected dominating set concept of  a
graph is applied.

 

4.3 . Conclusion

 

This
overview of dominating set in graph provides the idea usability of graph theory
and its applications in computer and communication networks. This above-compiled information related to the graph
theory and its applications in computer networks will provide useful
information for researchers to get some interest and  idea related to their field.