Vertex Minimal and Common Minimal Equitable Dominating Graphs

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 10, Vertex Minimal and Common Minimal Equitable Dominating Graphs G. Deepak a, N. D. Soner b and Anwar Alwardi b a Department of Mathematics The National Institute of Engineering, Mysore , India deepak1873@gmail.com b Department of Studies in Mathematics University of Mysore, Mysore , India ndsoner@yahoo.co.in Abstract In this paper we introduce the common minimal equitable and vertex minimal equitable dominating graph and we get characterize the common minimal equitable and vertex minimal equitable dominating graph which are either connected or complete, some new results of these graphs are obtained. Mathematics Subject Classification: 05C70 Keywords: Equitable common Minimal Dominating Graph, vertex minimal equitable dominating graph, equitable domination number 1 Introduction All the graphs considered here are assumed to be finite undirected with no loops and multiple edges. In general, we use X to denote the subgraph induced by the set of vertices X and N(v) and N[v] denote the open and closed neighbourhoods of a vertex v, respectively. A set D of vertices in a graph G is a dominating set if every vertex in V D is adjacent to some vertex in D. The domination number γ(g) is the minimum cardinality of a dominating set of G. A set S V is a neighbourhood set of G, ifg = v S N[v], where N[v] is the subgraph of G induced by v and all vertices adjacent to v. The neighbourhood number η(g) ofg is the minimum cardinality of a neighbourhood set of a graph G. A neighbourhood set S V is a minimal neighbourhood set, if S v for all v S, is not a neighbourhood set of G.

2 500 G. Deepak, N. D. Soner and A. Alwardi For terminology and notations not specifically defined here we refer reader to [3]. For more details about domination number and neighbourhood number and their related parameters, we refer to [4], [5], and [10]. A subset D of V (G) is called an equitable dominating set of a graph G if for every v (V D), there exists a vertex v D such that uv E(G) and d(u) d(v) 1. The minimum cardinality of such a dominating set is denoted by γ e and is called equitable domination number of G. D is minimal if for any vertex u D, D {u} is not a equitable dominating set of G. A subset S of V is called a equitable independent set, if for any u S, v/ N e (u), for all v S {u}. If a vertex u V be such that d(u) d(v) 2 for all v N(u) then u is an equitable dominating set. Such points are called equitable isolates. The equitable neighbourhood of u denoted by N e (u) is defined as N e (u) ={v N(u), d(u) d(v) 1}. The cardinality of N e (u) is denoted by d e (u). The maximum and minimum equitable degree of a point in G are denoted respectively by Δ e (G) and δ e (G). That is Δ e (G) =max u V (G) N e (u), δ e (G) =min u V (G) N e (u). Let S be a finite set and F = {S 1,S 2,..., S n } be a partition of S. Then the intersection graphs Ω(F )of F is the graph whose vertices are the subsets in F and in which two vertices S i and S j are adjacent if and only if S i S j φ. Kulli and Janakiram introduced many classes of intersection graphs in the field of domination theory see [5-8], also Anwar alwardi and N. D. Soner introduced some new intersection graphs by using the CN-domination see [1]. In this paper We define common minimal equitable dominating graph and vertex minimal equitable dominating graph,we obtain some fundamental and results of these graphs. 2 Common Minimal Equitable Dominating Graphs Definition 2.1. The common minimal equitable dominating graph is denoted by CED(G) is the graph which has the same vertex set as G with two vertices adjacent if and only if there exist minimal equitable dominating in G containing them. Example Let G as in Figure 1a. Then the minimal equitable dominating sets of G are A = {1, 4, 5}, B = {2, 4, 5} and C = {3, 4, 5} and the common minimal equitable dominating graph is shown in Figure 1b.

3 Vertex minimal and common minimal equitable dominating graphs Figure 1a Figure 1b Proposition 2.2. If G is complete graph, then CED(G) is totally disconnected. Proposition 2.3. If G is totally disconnected,then CED(G) is complete graph. Theorem 2.4. For any graph G, G CED(G) Proof. let u and v be any two adjacent vertices in G, then we can extend the set {u, v} into maximal equitable independent set S in G which is also minimal equitable dominating set that is u and v also adjacent vertices in CED(G). Hence G CED(G). Theorem 2.5. For any graph G, G = CED(G) if and only if every minimal equitable dominating set of G is independent. Proof. Let every minimal equitable dominating set of G is equitable independent. Then any two adjacent vertices in G can not be adjacent in CED(G), that is CED(G) G and by previous theorem, we get G = CED(G). Conversely, if CED(G) G, then any two vertices in the same minimal equitable dominating S of G are not adjacent in G. Hence S is independent. Let u V (G), the equitable neighbourhood of u denoted by N e (u) = {v N(u : d(u) d(v) 1}, the cardinality of N e (u) is denoted by d e (u), the equitable maximum degree Δ e (G) and the equitable minimum degree are defined respectively Δ e (G) =max u V (G) d e (u), δ e (G) =min u V (G) d e (u). Theorem 2.6. For any graph G with p vertices, where p 2, CED(G) is connected graph if and only if Δ e (G) <p 1. Proof. Let Δ e (G) <p 1 and u, v be any two vertices of G. Then we have four cases: Case 1: If u and v are not adjacent in G then by Theorem (2.4) u is adjacent to v in CED(G).

4 502 G. Deepak, N. D. Soner and A. Alwardi Case 2: If u and v are adjacent in G and there is a vertex w not adjacent to both u and v, then in CED(G), u and v are joining by the path uwv. Case 3: If u and v are adjacent in G and every other vertex w is adjacent and degree equitable to at least one of u and v. Then {u, v} is minimal equitable dominating set of G. Hence u is adjacent to v in CED(G). Case 4: If u and v are adjacent in G and there exist a vertex w adjacent to u or v but not equitable degree, then there exist two maximum equitable independent sets D 1 and D 2 contains u, w and v, w respectively then D 1 and D 2 are minimal equitable dominating set in G. Hence u and v are connected in CED(G) through w. From the four cases we get that CED(G) is connected graph. Conversely, suppose that CED(G) is connected graph. If possible suppose Δ e = p 1, then there exist at least one vertex u in G such that d e (u) =p 1, then u is isolated vertex in CED(G), and since G has at least two vertices implies that CED(G) has at least two component, a contradiction. Hence Δ e (G) <p 1. Proposition 2.7. If G with at least one isolated vertex, then CED(G) is complete graph. Proof. We know by Theorem in [2]that γ e (G) =p if and only if G has at least one equitable isolated point, that means there is only one minimal equitable domination set which contains all the vertices and by using the definition of CED(G) it is clear any two vertices are adjacent that means CED(G) is complete graph. Theorem 2.8. For any graph G with the property every equitable independent set in G is independent set, γ e (CED(G)) = p if and only if G is K p. Proof. If G is K p, then it is clear that the CED(G) is totally disconnected graph. Then γ e (CED(G)) = γ(ced(g)) = p. Conversely, suppose γ e (CED(G)) = p, then CED(G) is has at leastone equitable isolated. Then CED(G) =K p since all the minimal equitable dominating sets in G are equitable independent and by Theorem[2.5], G = K p. Hence G is K p. Conversely,if G is K p, then every minimal equitable dominating set of G is independent and by Theorem [2.5] CED(G) =G = K p. Hence γ e (CED(G)) = p. It is not true in general that if γ e (CED(G)) = p, then CED(G) is totally disconnected graph we show that by the following example Example. Let G be a graph as in Figure 2a, then the equitable minimal independent sets are {2}, {4}and{1, 3} and it is clear from Figure 2b that γ e (CED(G)) = p but CED(G) is not totally disconnected.

5 Vertex minimal and common minimal equitable dominating graphs Figure 2a Figure 2b 3 Vertex Minimal equitable Dominating Graphs Definition 3.1. The vertex minimal equitable dominating graph M v ED(G) of a graph G is a graph with V S as vertex set, where S is the collection of all minimal equitable dominating set of G with two vertices u, v V S are adjacent if they are adjacent in G or v = D is a minimal equitable dominating set of G containing u. Example. Let G be a graph as in Figure 1a. Then the minimal equitable dominating sets are A = {1, 4, 5}, B = {2, 4, 5} and C = {3, 4, 5}, and the vertex minimal equitable dominating graph M v ED(G) is shown in Figure 3. 1 B 2 3 A 1 1 C 4 5 Figure 3 Theorem 3.2. For any graph G, M v ED(G) is connected. Proof. Since for each vertex v V (G) there exist a minimal equitable dominating set containing v, every vertex in M v ED(G) is not isolated vertex. Now suppose M v ED(G) is disconnected, then there exist at least two component say G 1 and G 2 and there exist two nonadjacent vertices u, v such that

6 504 G. Deepak, N. D. Soner and A. Alwardi u G 1 and v G 2 that means there is no minimal equitable dominating set in G containing u and v, a contradiction. Hence M v ED(G) is connected. Theorem 3.3. For any graph G, diam(m v ED(G)) 3. Proof. Suppose G has at least two vertices. Then M v ED(G) has at least three vertices, let u, v V (M v ED(G)), we consider the following cases: Case 1: Suppose u, v V (G). Then in M v ED(G), d(u, v) 2. Case 2: Suppose that u V (G) and v / V (G). Then v = D is minimal equitable dominating set of G, ifu D, then in M v ED(G), d(u, v) = 1, if u/ D, then there exist vertex w D adjacent to u and has d(u) d(v) 1. Hence in M v ED(G) d(u, v) =d(u, w)+d(w, v) =2. Case 3: Suppose u, v / V (G). Then u = D and v = D are two minimal equitable dominating set in G, ifd and D are disjoint, then every vertex in D is adjacent to some vertex x D and vice versa this implies that in M v ED(G) d(u, v) =d(u, w)+d(w, x)+d(x, v) = 3, and if D and D are not disjoint then in M v ED(G), d(u, v) =d(u, w)+d(w, v) = 2,where w is common vertex between D and D. Hence diam(m v ED(G)) 3. Acknowledgments The authors are grateful to professor V. R. Kulli for his suggestions during the preparation of this paper. References [1] Anwar Alwardi and N. D. Soner, Minimal, Vertex Minimal And Common Minimal CN-Dominating Graphs,(manuscript). [2] K. D. Dharmalingam, Studies in Graph Theorey-Equitable domination and bottleneck domination, Ph.D Thesis (2006). [3] F. Harary, Graph theory, Addison-Wesley, Reading Mass (1969). [4] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of domination in graphs, Marcel Dekker, Inc., New York (1998). [5] S. M. Hedetneimi, S. T. Hedetneimi, R. C. Laskar, L. Markus and P. J. Slater. Disjoint dominating sets in graphs. Proc. Int. Conf. on Disc.Math., IMI-IISc, Bangalore (2006) [6] V. R. Kulli and B. Janakiram, The Minimal Dominating Graph, Graph Theory Notes of New York, New York Academy of Sciences, 28(1995),

7 Vertex minimal and common minimal equitable dominating graphs 505 [7] V. R. Kulli, B. Janakiram and K. M. Niranjan, The Common Minimal Dominating Graph, Indian J. Pure. appl. Math., 27(1996), [8] V. R. Kulli, B. Janakiram and K. M. Niranjan, The Vertex Minimal Dominating Graph, Acta Ciencia Indica, 28(2002), [9] V. R. Kulli, B. Janakiram and K. M. Niranjan, The Dominating Graph, Graph Theory Notes of New York, New York Academy of Sciences, 46(2004), 5-8. [10] H. B. Walikar, B. D. Acharya and E. Sampathkumar, Recent developmentsin the theory of domination in graphs, Mehta Research Institute, Allahabad, MRI Lecture Notes in Math. 1 (1979). Received: September, 2011