Domination and Irredundant Number of 4-Regular Graph

Transcription

1 Domination and Irredundant Number of 4-Regular Graph S. Delbin Prema #1 and C. Jayasekaran *2 # Department of Mathematics, RVS Technical Campus-Coimbatore, Coimbatore , Tamil Nadu, India * Department of Mathematics, Pioneer Kumaraswamy College, Nagercoil , Tamil Nadu, India 1 delbin.prema@gmail.com, 2 jaya_pkc@yahoo.com Abstract - A set D of vertices in a graph G is a dominating set of G if every vertex V (G) \ D is adjacent to some vertex in D. The minimum cardinality of a dominating set of G is the domination number of G denoted as (G). A set D is irredundant if for every vertex u D, we have N [u] N [D {u}]. A subset D V is irredundant if for each x D, D {x} does not dominate N [x]. Equivalently, D is an irredundant set of vertices if N [D {v}] N [D] for every vertex v D. These parameters are examined, especially how they relate to one another and to other graph parameters and their behaviour in certain graph classes. Keywords - Regular graph, Domination set, Domination number, irredundant set, irredundant number. I. INTRODUCTION Graph theory is one of the most important branches of modern mathematics and computer applications. The past 40 years have witnessed spectacular growth of Graph theory due to its wide application to discrete optimization problems, combinatorial problems and classical algebraic problems. It has a very wide range of applications to many fields like engineering, physical, social and biological sciences, linguistics etc., the theory of domination has been the nucleus of research activity in graph theory in recent times. This is largely due to a variety of new parameters that can be developed from the basic definition of domination. In particular, the notation of irredundancy is relevant in the context of communication networks, since any irredundant node in a set can be removed from the set without affecting the totality of nodes that may receive communication from node in the set. The concept of irredundance was introduced originally in [3]. The literature on this subject has been surveyed and detailed in the two books by Haynes, Hedetniemi and Slater [5, 6]. It is closely related to domination and irredundance in graph theory. An excellent bibliography of existing results is given in [4]. In this paper we discuss the relation of domination and irredundance by 4-regular graph. For a graph G = (V, E) and a vertex v V. The open neighbourhood of v is the set N (v) = {u V uv E} and the closed neighbourhood of v is N [v] = N (v) {v}. If u N [v], we also say that v dominates u and if U = V W N [v], then W dominates U. If U = V, then W is said to dominate G or to be a dominating set of G. It was specified in [8]. A dominating set is a set w V for which N [w] = V, or equivalently, for every vertex u V D, we have N (u) D. A set D of vertices in a graph G is a dominating set of G if every vertex in V D is adjacent to some vertex in D. A vertex v in a graph G is said to dominate itself and each of its neighbours. 237

2 We say with other words, v dominates the vertices of its closed neighbourhood N [v]. The domination number of G equals the minimum cardinality over all dominating set in G and it is denoted by (G). A dominating set of cardinality (G) is referred to as a minimum dominating set. A dominating set of cardinality (G) is called a -set. That are explained in [7, 9]. A minimal dominating set in a graph G is a dominating set that contains no dominating set as a proper subset. A minimal dominating set of minimum cardinality is a minimum dominating set and consists of (G) vertices. A set D is irredundant if for every vertex u D, we have N [u] N [D {u}]. A subset D V is irredundant if for each x D, D {x} does not dominate N [x]. Equivalently, D is an irredundant set of vertices if N [D {v}] N [D] for every vertex v D. The irredundant number of G, denoted by ir(g), is the minimum cardinality taken over all maximal irredundant set of vertices of G. An irredundant set of cardinality ir(g) is called an ir-set. It was specified in [2, 10]. A set D V is a dominating set of G if N [D] = v, while D V is an irredundant set of G if PN (x, D) = N [x] N [D {x}] for every x D. D is a minimal dominating if and only if it is dominating and irredundant. An irredundant set D is maximal irredundant if no proper superset of D is irredundant. Which are indicated in [1, 11]. Every vertex v with the property N [D {v}] N [D] is an irredundant vertex. Consequently, every vertex in an irredundant set is an irredundant vertex. II. DOMINATION AND IRREDUNDANT PARAMETERS Motivated by the study of domination and irredundance, we have investigated some of the theorems based on domination number and irredundance number. Theorem 2.1: If G is a graph then ir(g) = n/2 1. Proof: We must first complete the basis step, i.e., we have to show that 1. Basis of induction: For n =1 we have ir(g) = 6/2(4) 1 = 6/7 = 0.8 = 1 2. Induction step: Assume that ir(g) = k/2(4) 1 = k /7 we have ir(g) = k + 1/2(4) 1 = k + 1/7 Therefore by the principle of mathematical induction, the result is true for all n N. 238

3 Theorem 2.2: A dominating set D of a graph G is a minimal dominating set of G iff every vertex v in D satisfies atleast one of the following two concepts: i) there exists a vertex w in V (G) D such that (G) D ={v} ii) V is adjacent to no vertex of D. Proof: If each vertex v in D has atleast one of the concepts then D {v} is not a dominating set of G. Consequently, D is a minimal dominating set of G. Conversely, assume that D is a minimal dominating set of G. Then for each v D, the set D-{v} is not a dominating set of G. Hence there is a vertex w in V (G) (D {v}) that is adjacent to no vertex of D {v}. If w = v then v is adjacent to no vertex of D. Suppose that w v. Since D is a dominating set and w D, the vertex w is adjacent at least one vertex of D. However, w is adjacent to no vertex of D {v}. Consequently, (w) D = {v}. Theorem 2.3: If D is a minimal dominating set of a graph G, then V (G) - D is a dominating set of G. Proof: Let v D. By Theorem 2.2, every vertex v in D either v dominates some vertex of V (G) - D such that (G) D = {v} or no other vertex of D dominates v. Suppose first that there exists a vertex w in V (G) - D such that (G) D = {v}. Hence v is adjacent to some vertex in V (G) - D. Suppose next that v is adjacent to no vertex in D. Then v is an isolated vertex of the subgraph < D >. Since v is not isolated in G, the vertex v is adjacent to some vertex of V (G) - D. Thus V (G) - D is a dominating set of G. Theorem 2.4: If G is a graph of order n then γ(g) < n/2. Proof: Let D be a minimal dominating set of G. By Theorem 2.3, if D is a minimal dominating set of a graph G, then V (G) - D is a dominating set of G. Thus γ(g) < min{ D, V (G) D } < n/2. Theorem 2.5: If G is a graph and D V (G) is a dominating set then D is an irredundant set. Proof: Let w D and we may assume that w is not an adjacent to any vertex V - D, then by dominance for every vertex w D, N(w) D =. i.e., every vertex w D is not adjacent to any vertex V - D. Our assumption is controdiction to the dominating set. Therefore N(w) D. Every vertex in D is adjacent to atleast any one vertex in V - D. Observe that each such w is an element of N[w] - N[D {w}]. i.e., I(w,D) = {w, z}. Therefore I(w, D). Hence D is an irredundant set. 239

4 Theorem 2.6: If D is a dominating set of a graph G then ir(g) = (G). Proof: Let the vertices of a graph G be V (G) = {u, v, w, x, y, z}. Then the dominating set of G is D = {w, z}. Here the dominating number (G) = 2. Next we should prove that the irredundance number ir(g) = 2. Let us consider that N[w] = {u, v, w, x, y}. D {w} = {z}. Then the neighbour of D {w} is {u, v, z, x, y}. N[w] - N[D {w}] = {w, z}. Therefore ir(g) = 2. Here both ir(g) and (G) are equal. i.e., the irredundant number of graph G is equal to the domination number of a graph G. Hence ir(g) = (G). Theorem 2.7: If a irredundant set is defined for a graph G then a minimal dominating set D is a maximal irredundant set. Proof: Suppose D is a minimal dominating set. Then for w D either D {w} is not dominating set or the induced graph < (u \ D) {w} > is connected. By Theorem 2.5, follows that D is an irredundant set. Now we want to show that D is a maximal irredundant set. Let v be an arbitrary vertex in V\D, Since D is a minimal dominating set, v N[D] and hence, D {v} is not an irredundant. For every v V \ D, v has not a private neighbour with respect to V (D {v}) or < V \ {D v} > is connected. As a result, D is a maximal irredundant set. III. CONCLUSION In this paper we discussed the relation between domination and irredundance number in 4-regular graph. Domination and irredundance can stand together to facilitate the network communication process. They play very vital role in coding theory, computer science, operation research, switching circuits, electrical network, etc. REFERENCES [1] Michael A. Henning, Irredundance perfect graphs, Discrete Math., vol. 142, pp , July [2] S. Delbin Prema and C. Jayasekaran, The Detour irredundant number of a graph, International Journal of Pure and Applied Mathematics, 11 Aug 2017 (Accepted). [3] E.J. Cockayne, S.T. Hedetniemi and D.J. Miller, Properties of hereditary hypergraphs and middle graphs, Canad. Math. Bull., vol. 21, pp , Dec [4] S.T. Hedetniemi, R. Laskar and J. Pfaff, Irredundance in Graphs - A Survey, Clemson Univ., Computer Science Dept., Internal Research Report. 240

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