Literature one of the immensely researched branches is domination

LiteratureReview Any real-worldsituation can be interpreted graphicallywith a group of points joined with lines.For example, a computer network can be represented graphically in which thepoints represent computer terminals and lines representing communication linksbetween computer. A mathematical abstraction of real-worldsituations which focuses on representing situation by using points,which are connected together give rise to the concept called graph 36.

Dueto 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 researchedareas of modern mathematics which has seen an exponential growth. In graphtheory, one of the immensely researched branches is domination in the graph 7.  In the last threedecades, an exponential growth has been seen in graph theory because of itswide range of applications in optimization problems, combinatorial problems,classical algebraic problems, computational problems etc. This growth is mainlydue to the rise of a number of new parameters that have been developed from the basic definition of domination.  The intense research ofdominating set in graph theory has been started around the early 60’s. Thisbranch of graph theory has its historical roots during 1862 when C.F. DeJaenisch started studying the problem of chessboard game moves in which hetried determining the minimum number of queens that are needed to dominate an n × n chessboard.

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   Soon in coming years,W.W. Rouse Ball also reported three basic chess problem namely; coveringproblem, 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 dominationnumber of a graph.  In the year 1962, Ore has used the termsdominating set and domination number for the same concept in graph theory 811.  The graph theorists E.J.

Cockayne,S.T. Hedetniemi, R.C. Laskar have made some of the interesting and broadreviews on the results of dominating sets in graphs during 70’s.   Othergraph 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 asignificant research in dominating set, domination numbers 13 15 and otherrelated topics in graph theory.  Theapplications of dominating set and domination number of   the graph in computer networking  will be discussed in detail in the followingsection.

 4.2 Domination   Sets in ComputerNetworks Dominationis an area in graph theory with an extensive research activity. The dominatingset problem tells to determine the domination number of a graph. The concept ofdominating set occurs in a variety of problems. A number of problems aremotivated by communication network problems. The communication network includesa set of nodes, where one node can communicate with another, if it is directlyconnected to that node. In order to send a message directly from a set of nodesto all others, one has to choose the set in such a way that all other nodes areconnected to at least one node in the set.

Now, such a set is a dominating setin a graph, which represents the network .Other applications of domination arethe facility location problem, land surveying and routings. Its applicationlies in various fields of solving mathematical abstraction of real-lifeproblems Social Network Theory  A social networking service is anonline platform that people use to build social networks or social relations with other people whoshare similar personal or career interests, activities, backgrounds orreal-life connections.

 Theonline social network has been developed significantly and immensely grown inthe recent years as a medium of communication, sharing the information andspreading the influence 1. The data from the social network can be used tocarry out research on the various social problemand can be utilized in disseminating the information and ideas. The online social network can be utilized forsolving general physiological and social problems in the physical world such asdrinking, smoking, and drug problems are all explored well.   Thedominating set plays a vital role in analyzing the effect on a real onlinesocial network data set throughsimulation.  The dominating set concept can beapplied to the social network by stimulating it as a graph in which pointrepresent user and links are used to represent relationship, this network graphcan be used to determine the amount of positive or negative influence that ispossessed by an individual as well as its impact on their related neighbor. This degree threshold positivity spreadby a dominating positive user can spread the positive educational influencethroughout the entire community in the social network.

 The positive, as well as negative impactof different people in a social setup, can play different roles as they areaffected by their peers. The positive or negative influence in social issuescan move in two directions. That is, a positive individual can convert into anegative individual and can move back and forth between these two states formultiple times, that is they can change their state by the influence of theirneighbor. An undirected graph G = (V, E, C) isused to denote the online social network, because friendship in an onlinesocial network is usually bidirectional as shown in Figure 4.21. Fig 4.21 Social Network Architecture Instance Descriptionof the above figure: Online social network can be represented as a graph ofrelationship with individuals representing the nodes of a graph (V), the socialinteractions as edges (E) and C is the compartment vector that saves thecompartment of each node. The compartment part of a node decides whether thesocial issue(s) of an individual has a positiveor negative impact on its neighbors.

 Computer Communication Networks  Thedominating set plays a vital role in networking and information exchange in acomputer and communication networks to route the correct information betweenthe receiver and sending nodes. For example, wireless network. A wirelessnetwork is a type of computer network that uses wireless data transmission byconnecting the wireless network nodes. The examples of a wireless network are cell-phone network, mobileAd-hoc network, wireless sensor network etc.  Dominating Set in Mobile Ad-hoc Network (MANET) AMobile Ad-Hoc Network (MANET) is a self-configurable infrastructure-less network which is used for connecting the mobiledevices with accurate transmission link in wireless mode.

Routing andbroadcasting the information between mobile devices in mobile ad-hoc networksis studied and implemented using dominating set 18.  The Connected Dominating Set (CDS) is widelyused as a virtual backbone for MANETs to provide different communicationprimitive such as routing, broadcasting etc 14. The construction of virtualbackbone is very important and it is approximated by a minimum CDS with a unitdisk graph 4 as shown in Figure 4.22 Fig. 4.22 CDS Construction in MANETs Theconnected dominating set is defined as a subset of nodes in a wireless networksuch that each node in the network is either present in the set or as a neighbor of some node in the set and theinduced graph with the nodes in the set is connected. The CDS is useful inreducing the communication and the storage overhead by keeping its size to beminimal.

However, it is well–known that the problem of finding the minimalCDS is     NP–hard 19 and it can be solved practically through preprocessing2. Dominating Set in Wireless Sensor Network   A Wireless Sensor Network (WSN) is atype of wireless network which consists of spatially distributed autonomoussensors to monitor the physical or environmental conditions such astemperature, pressure, sound etc. and to broadcast their information throughthe wireless network to a central location 19 20.   The main activity in the WSN thus formsthe timely as well as correct routing of information between the nodes. Routingin WSN is a very challenging because of the inherent characteristics ofdistinguishing these networks from other wireless networks such as MANETs andother cellular networks.

A number of routing techniques are available forwireless sensor networks. Among them, hierarchical routing and cluster-basedrouting methods are well-known techniquesfor scalable and efficient communication. The hierarchical routing with virtualbackbone infrastructure is mainly employed to perform energy–efficient routingin WSN(s). Thus, the CDS comes into the pictureto serve as a virtual backbone for WSNs to reduce the routing overheads. TheCDS incorporated hierarchical techniques simplifies the routing by restrictingthe main routing task to the dominating nodes only.

 Minimal Communication   power consumption in Wireless Sensor Network  Agroup of wireless sensors forms a wireless sensor network to become a veryresponsive and broadly operating network. In a wireless network along with itsresponsiveness and time consumption amount of power,use plays a vital role in its credibility. A virtual backbone routingenvironment of a wireless network consists of a connectedsubset of nodes whichis responsible for routing the information in the network. A node in thatsubset is likely to get exhausted because of performing heavy duties ascompared to other nodes. This situation can get more intensified if the nodeuses high communication power to form the virtual backbone.

Hence to compute avirtual backbone in a wireless network with minimal total communication powerthe connected dominating set concept of  agraph is applied.  4.3 .

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