APplication Domination in Graph

Volume : 2 | Issue : 3 | March 2013 ISSN - 2250-1991Research Paper Engineering Domination in Graph with Application * Preeti Gupta * Department of Engg. Mathematics, Prestige Institute of Engineering and Science, Indore, INDIA ABSTRACT The paper concentrates on the domination in graphs with application In a graph G = (V, E ), S ⊆ V is a dominating set of G if every vertex is either in S or joined by an edge to some vertex in S. Many different types of domination have been researched extensively this paper explores applications of dominating sets. 2000 Mathematics Subject classification: 05C69, 05C99 Keywords: Domination Set. 1. INTRODUCTION ingful sub areas, placing the study of dominating sets in even Domination in graphs has been an extensively researched broader mathematical and algorithmic contexts. Vasumathi & branch of graph theory. Graph theory is one of the most Vangipuram[8] and Vijayasaradhi & Vangipuram [9] obtained flourishing branches of modern mathematics and computer domination parameters of an arithmetic Graph and also they applications. The last 30 years have witnessed spectacular have obtained an elegant method for the construction of a growth of Graph theory due to its wide applications to discrete arithmetic graph with the given domination parameter. In most optimization problems, combinatorial problems and classical of the researches in Graph theory, the investigators are con- algebraic problems. It has a very wide range of applications tent with establishing the existence of a graph with a given to many fields like engineering, physical, social and biologi- graphical parameter. For example, given domination number cal sciences; linguistics etc., the theory of domination has as n does there exist a graph with this as the domination num- been the nucleus of research activity in graph theory in re- ber? Similarly does there exist a graph with given bondage cent times. This is largely due to a variety of new parameters number or with given domatic number? These problems have that can be developed from the basic definition of domina- been investigated successfully. However in the matter of ap- tion. The NP-completeness other basic domination problems plications of these results to real life situations it becomes and its close relationship to other NP-completeness problems necessary to evolve the method of constructing such a graph have contributed to the enormous growth of research activity with a given parameter. Construction of a graph with a given in domination theory. It is clearly established from the exclu- Graph theoretic parameter is generally difficult by the usual sive coverage of the “Topics on domination in graph” in the graph theoretic methods. In many applications of domination 86th issue of the Journal of Discrete mathematics (1990), number, bondage number, or domatic number, it becomes that the theory of domination is a very popular area for re- necessary to construct a graph with as few vertices and/or search activity in graph theory. The rigorous study of domi- edges as possible with a given domination number or bond- nating sets in graph theory began around 1960, even though age number or domatic number. It is in this context the us- the subject has historical roots dating back to 1862 when de age of elementary number theoretic principles will help in the Jaenisch studied the problems of determining the minimum constructions of such graphs. In Vasumathi and Vangipuram number of queens which are necessary to cover or dominate [8], the construction of a graph with a given domination num- a n x n chessboard. In 1958, Berge defined the concept of ber has been given, using such a method. A similar method the domination number of a graph, calling this as “coefficient of construction using again elementary principles of number of External Stability”. In 1962, Ore used the name “dominat- theory helped in the construction of a graph with a graceful ing set‟ and “domination number‟ for the same concept. In degree sequence by Vijayasaradhi and Vangipuram [9]. 1977 Cockayne and Hedetniemi made an interesting and ex- tensive survey of the results know at that time about dominat- 2. Applications of Domination in Graph ing sets in graphs. They have used the notation (G) for the Domination in graphs has applications to several fields. Domi- domination number of a graph, which has become very popu- nation arises in facility location problems, where the number lar since then. The survey paper of Cockayane and Hedet- of facilities (e.g., hospitals, fire stations) is fixed and one at- niemi has generated lot of interest in the study of domination tempts to minimize the distance that a person needs to travel in graphs. In a span of about twenty years after the survey, to get to the closest facility. A similar problem occurs when more than 1,200 research papers have been published on the maximum distance to a fality is fixed and one attempts to this topic, and the number of papers continued to be on then minimize the number of facilities necessary so that everyone crease. Since then a number of graph theorists Konig, Ore, is serviced. Concepts from domination also appear in prob- Bauer, Harary, Lasker, Berge, Cockayne, Hedetniemi, Alavi, lems involving finding sets of representatives, in monitoring Allan, Chartrand, Kulli, Sampthkumar, Walikar, Armugam, communication or electrical networks, and in land surveying Acharya, Neeralgi, Nagaraja Rao, Vangipuram many others (e.g., minimizing the number of places a surveyor must stand have done very interesting and significant work in the domi- in order to take height measurements for an entire region). nation numbers and the other related topics. Recent book on domination [3], has stimulated sufficient inspiration leading 2.1 School Bus Routing: to the expansive growth of this field of study. It has also put Most school in the country provide school buses for transport- some order into this huge collection of research papers, and ing children to and from school Most also operate under cer- organized the study of dominating sets in graphs into mean- tain rules, one of which usually states that no child shall have PARIPEX - INDIAN JOURNAL OF RESEARCH X 115 The goal is to locate a radar for the surveillance at as few of these locations as possible. What is the least number of stations in a set which dominates (within distance 50) all other vertices in this graph? A set (B. 2. Using graph theory as a modeling tool in biological networks allows the utilization of the most graphical invariants in such Let each village be represented by a vertex. Each processor to which it is directly connected. How a set of locations in which the radar stations are to be placed can be determined. Since this must be done edges in the graph. We would like to locate radio stations node in the subset.2 Computer Communication Networks: Consider a computer network modeled by a graph G = (V. domination is considered to be one of the funda- mental concepts in graph theory and its various applications to ad hoc networks. Let us say that the following figure represents a street map of part of a city. There are various locations and an arc can be drawn from lo- cation x to location y if it is possible for a watchman stationed at x to observe a warning light located at y. 27. In this case we seek a distance-2 dominating set among the set 2. of 16 com- puters. Let us assume that the school has decided that no child shall have to walk more than two blocks in order to be picked up by a school bus. or network. social networks and web graphs [1. A number of strategic dominating set in the hypercube network in following figure locations are to be kept under surveillance. range of only fifty kilometers. 47] partly explain the increased interest. An edge between a way that it is possible to identify secondary RNA (Ribonu- two villages is labeled with the distance. they must construct a route for each bus that gets with- in one quarter km of every child in its assigned area. we can essentially remove all ing processors (a dominating set). and where should they be located? At present. The school is located at the large vertex. But since radio stations are costly. gets within two blocks of every child and returns to the school. Let us assume that a radio station has a broadcast range of fifty kilometers.Volume : 2 | Issue : 3 | March 2013 ISSN . Notice that if we could afford radio stations which have close to all other processors. say one quarter km to a bus pickup point.3 Radio Stations: to select a subset of nodes that will provide some definite Suppose that we have a collection of small villages in a re. for which vertices represents computers and edges represent direct links between pairs of computers. a limited broadcasting range. say in kilometers. where each edge represents one pick up block. Let us say that we will tolerate a broadcast range of seventy kilometers.E). How many guards are needed to observe all of the warning lights. 2. F.2250-1991 to walk farther than. The two shaded vertices form a distance-2 The problem was discussed by Berge . Construct a route for a school bus that leaves the school.INDIAN JOURNAL OF RESEARCH . or processors. The results 116 X PARIPEX . we must use several stations to reach all villages. Let the vertices in following figure represent an array. Those graphical invariants are between the two villages variations of the domination number of a graph. we want 2. which represent a distance of more than relatively fast. Since each radio station has problems.4 Locating Radar Stations Problem of all processors. service such that every node in the network is ‘close’ to some mote part of the world.J} of cardinality four is indicated in the following figure(b). We need only to find a dominating set in this a path. Assume that from time to time we need to collect information from all processors. cleic acid) motifs numerically. No bus ride can take more than some specified number of minutes. three radio stations at most a two unit delay between the time a processor sends would sufficient. its information and the time it arrives at a nearby collector. Thus. 25. distributed comput- ing. The following examples show when the in some of these villages so that messages can be broadcast concept of domination can be applied in modeling real-life to all of the villages in the region. Thus we identify a small set of processors which are graph. Such applications usually aim 2. H. biological networks. we cannot route this information over too long fifty kilometers.6 Modeling Biological Networks to locate as few as possible which can reach all other villages.5 Nuclear Power Plants Problem z similar known problem is a nuclear power plants problem. We do this by having each Here we have assumed that a radio station has a broadcast processor route its information to one of a small set of collect. and Limits on the number of children that a bus can carry at any one time. . Fundamentals of domination in graphs.. social networks are modeled in terms of graph tions. p)- among the trees that represent native structures and those covering sets..J.g. F. Graphs: An Introductory Approach.C. Proceedings of a symposium on graph theory and combinatorics. 308-320. 141-147. Networks.T. The hierarchical overlay networks usually serve choice of initial sets of target individuals is an important prob.10 Multiple Domination Problems studying the dynamics of relations among numerous individu. E.7 Modeling Social Networks Dominating sets can be used in modeling social networks and 2. Wilson and J. p > 1. then the sets of vectors which are (n. p}. 33-36. Mas- sachusetts. Facility loca- tion problems are concerned with the location of one or more 3. So. Domination of undirected graphs. | [11] Cockayne. Marcel Dekkar. E. Researches may get some information related and Hedetniemi .W. (1969).J. If one defines a graph.W.. Vol. Existence of a graph with a given domination parameter | Proceedings of the Fourth Ramanujan Symposium on Algebra and its Applications.... An overview is present- 2. Inc-New York (1998). Addison – Wesley. G. important and quickly developing applications of 2. S. Wiley & Sons 1990. e. Graph Theory: An Introductory Course Springer 1979. construction of minimum spanning trees.T. single error correcting codes.. On domination related concepts in graph theory. ity location problems in operational research. J. | [9] Vijaya Saradhi and Vangipuram: „Irregular graphs‟.Volume : 2 | Issue : 3 | March 2013 ISSN . the graph The concept of domination is also applied in coding theory as theory section of each paper is given importance than to the discussed by Kalbfleisch. On some new domination parameters of a graph – A Survey. T. 2. pp. University of Madras. R. S. in modern lem in the theory of social networks. which are con. | [2] Bondy and Murty: Graph theory with applications. 7-13. (1980). S. towards a theory of domination in graphs. | [12] Cockayne.. Watkins John.T. Kochi. Stanton and Horton and Cockayne other sections.B.. Networks. Chapman and Hall. 2001.. 10.. Graph Theory. providing equitable service to graph theoretical ideas in various areas of Science & Engi- customers and capturing the largest market share. A survey. and Hedetniemi.. Graphs and digraphs. Conclusion: facilities in a way that optimizes a certain objective such as The main aim of this paper is to present the importance of minimizing transportation cost. | [4] Harary. Lecture notes in Math. and Hedetniemi.. N. In the work of Kelleher file sharing and instant messaging computer network applica- and Cozzens.nates chosen from get some ideas related to their field of research. 247-271.. or perfect cover- that are not likely candidates to represent RNA.J.B. neering for researches that they can use Domination in graph theoretical concepts for the research. | [3] Haynes. and Walikar. ing sets are all dominating sets of the graph with determined additional properties. E. Dominating sets of several kinds are used for balanc- theory and it was shown that some of these sets can be found ing efficiency and fault tolerance as well as in the distributed by using the properties of dominating sets in graphs. H..9 Coding Theory ed especially to project the idea of graph theory. and Lesniak. Hedetniemi.8 Facility Location Problems multiple domination in modern computer networks is a wire- The dominating sets in graphs are natural models for facil. pp. | [10] Chatrand.J.. networks in peer-to-peer applications for more efficient index nected by one or more specific types of interdependency. 642 (1978). | [13] Cockayne. India. Kerala. R. in : Lecture in Match. 17-19 may 1991. and Vangipuram. Madras (1996). 187-195 (1995).. 885 (1981). Dawes. | [5]..INDIAN JOURNAL OF RESEARCH X 117 . | [7] Sampathkumar. Total domination in graphs.J. Macmillan (1976). and two vertices are adjacent if they differ domination number can be used for correctly distinguishing in one coordinate. L. A social network is a social structure domination can be used to construct hierarchical overlay made of individuals (or groups of individuals).2250-1991 of the research carried out in show that the variations of the {1. PARIPEX . the vertices of which to graph theory and its applications in various field and can are the n-dimensional vectors with coordi. P. | [8] Vasumathi. REFERENCES [1] Bollobas. fall (1977). and Slater. Graph Theory Notes of New York. S. less sensor network. E. The searching. | [6] Laskar. as distributed databases for index searching. Multiple als in different domains.. Another good exam- ple of direct. Madras. An important role is played by multiple domination. 41. 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