Restrained Weakly Connected Independent Domination in the Corona and Composition of Graphs

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1 Applied Mathematical Sciences, Vol. 9, 2015, no. 20, HIKARI Ltd, Restrained Weakly Connected Independent Domination in the Corona and Composition of Graphs Rene E. Leonida Mathematics Department College of Natural Sciences and Mathematics Mindanao State University Fatima, General Santos City, Philippines Copyright c 2015 Rene E. Leonida. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we explore the concept of restrained weakly connected independent domination in graphs. In particular, we characterized the restrained weakly connected independent dominating sets in the corona, and composition of graphs and; as a consequence, their restrained weakly connected independent domination numbers are obtained. Mathematics Subject Classification: 05C69 Keywords: domination, restrained domination, independent domination, weakly connected domination, restrained weakly connected independent domination 1 Introduction and Preliminary Results Let G = (V (G), E(G)) be a simple connected graph. For any vertex v V (G), the open neighborhood of v is the set N(v) = {u V (G) : uv E(G)} and the closed neighborhood of v is the set N[v] = N(v) {v}. For a set X V (G), the open neighborhood of X is N(X) = v X N(v) and the closed neighborhood of X is N[X] = X N(X). A subset S of V (G) is an

2 974 Rene E. Leonida independent set if for every x, y S, xy / E(G). The independence number β(g) of G is the largest cardinality of an independent set of G. A subset S of V (G) is called weakly connected if the subgraph S w = (N G [S], E W ) weakly induced by S, is connected, where E W is the set of all edges with at least one vertex in S. A subset S of V (G) is a dominating set of G if for every v V (G)\S, there exists u S such that uv E(G). The domination number γ(g) of G is the smallest cardinality of a dominating set of G. A dominating set of G which is independent is called an independent dominating set of G. The independent domination number i(g) of G is the smallest cardinality of an independent dominating set of G. A dominating set of G which is weakly connected is called a weakly connected dominating set. The weakly connected domination number γ w (G) of G is the smallest cardinality of a weakly connected dominating set of G. An independent dominating set of G which is weakly connected is called a weakly connected independent dominating set. The weakly connected independent domination number i w (G) of G is the smallest cardinality of a weakly connected independent dominating set of G. Similarly, the upper weakly connected independent domination number β w (G) of G is the largest cardinality of a weakly connected independent dominating set of G. A dominating set S is called a restrained dominating set of G if for every u V (G)\S, there exists w V (G)\S such that uw E(G). The restrained domination number of G, denoted by γ r (G), is the smallest cardinality of a secure dominating set of G. A set S is called a restrained weakly connected independent dominating set of G if S is a weakly connected independent dominating set of G and for every u V (G)\S, there exists w V (G)\S such that uw E(G). The restrained weakly connected independent domination number of G, denoted by i rw (G), is the smallest cardinality of a restrained weakly connected dominating set of G. The concept of weakly connected independent domination is discussed in [3] [4], and [5]. Another domination parameter is the restrained domination which was discussed in [1], [2], and [5]. A combination of these two concepts give rise to a new variant of domination called restrained weakly connected independent domination. 2 Corona of Graphs Let G and H be graphs of order m and n, respectively. The corona G H of G and H is the graph obtained by taking one copy of G and m copies of H, and then joining the ith vertex of G to every vertex of the ith copy of H. For every v V (G), denote by H v the copy of H whose vertices are attached one by one to the vertex v. Denote by v + H v the subgraph of the corona G H corresponding to the join {v} + H v.

3 Restrained weakly connected independent domination 975 The following theorem will be useful. Theorem 2.1 Let G be a connected graph of order n 3 and let K 1 = {v}. Then S V (K 1 +G) is a restrained weakly connected independent dominating set of K 1 + G if and only if one of the following holds: (i) S = {v}. (ii) S is an independent dominating set of G. Proof : Suppose S V (K 1 + G) is a restrained weakly connected independent dominating set of K 1 +G. If v / S, then S V (G). Clearly S is a dominating set of G. For the converse, if S = {v}, then S is a restrained weakly connected independent dominating of K 1 + G. Suppose S is an independent dominating set of G. Then S is a weakly connected independent dominating set of K 1 +G. Since vx E(K 1 + G) for all x V (G)\S, it follows that S is a restrained weakly connected independent dominating set of K 1 + G. The following result characterizes the restrained weakly connected independent domination in the corona of two connected graphs. Theorem 2.2 Let G and H be connected graphs of order m 2 and n 3, respectively. Then S V (G H) is a restrained weakly connected independent dominating set of G H if and only if S = S 1, where S 1 is v V (G)\S 1 S v a weakly connected independent dominating set of G and S v is an independent dominating set of H v for all v V (G)\S 1. Proof : Suppose S V (G H) be a restrained weakly connected independent dominating set of G H. Let S 1 = S V (G). Since S is a weakly connected independent dominating set of G, S 1 is a weakly connected independent dominating set of G. Let v V (G)\S 1. By Theorem 2.1, S v is an independent dominating set of H v. Hence, S = S 1. Conversely, suppose S = S 1 v V (G)\S 1 S v v V (G)\S 1 S v, where S 1 is a weakly connected independent dominating set of G and S v is an independent dominating set of H v for all v V (G)\S 1. By Theorem 2.1, {v} is a restrained weakly connected independent dominating set of v + H v for each v S 1 and S v is a restrained weakly connected independent dominating set of v + H v for each v / S 1. Therefore, S is a restrained weakly connected independent dominating set of G H.

4 976 Rene E. Leonida The following theorem can be found in [4]. Theorem 2.3 Let G be a connected graph of order m and H any graph with i(h) 1. If C V (G H) is a minimum weakly connected independent dominating set of G H, then C V (G) is a maximum weakly connected independent dominating set of G. Corollary 2.4 Let G and H be connected graphs of order m 2 and n 3, respectively. Then i rw (G H) = β w (G) + (m β w (G))i(H). Proof : The corollary clearly holds when i(h) = 1. Suppose i(h) 1. Let S 1 be a maximum weakly connected independent dominating set of G and S be a minimum independent dominating set of H. For each v V (G)\S 1, let S v V (H v ) be such that S v = S. Let S 2 = {S v : v V (G)\S 1 }. By Theorem 2.2, S = S 1 S 2 is a restrained weakly connected independent dominating set of G H. Thus, i rw (G H) S = S 1 + v V (G)\S 1 S v = β w (G) + (m β w (G))i(H). Next, let S be a minimum restrained weakly connnected independent dominating set of G H. Let S 1 = S V (G) and S 2 = S \S 1. For each u V (G)\S 1, let S u V (H u ) be an independent dominating set of H u. Then S 2 = {S u : u V (G)\S 1 }. By Theorem 2.3, S 1 is a maximum weakly connected independent dominating set of G. Thus, S 1 = β w (G). Hence, i rw (G H) = S = S 1 + u V (G)\S 1 S u β w (G) + (m β w (G))i(H). Therefore, i rw (G H) = β w (G) + (m β w (G))i(H). 3 Composition of Graphs Observe that a subset C of V (G[H]) = V (G) V (H) can be written as C = ({x} T x ), where S V (G) and T x V (H) for every x S. Henceforth, we shall use this form to denote any subset C of V (G[H]). The following result can be found in [5]. Theorem 3.1 Let G be a nontrivial connected graph and H any graph. A subset C = ({x} T x ) of V (G[H]) is a weakly connected independent dominating set of G[H] if and only if S is a weakly connected independent dominating set of G and T x is an independent dominating set of H for every x S.

5 Restrained weakly connected independent domination 977 A similar result characterizes the restrained weakly connected independent dominating set of G[H]. Theorem 3.2 Let G and H be nontrivial connected graphs. A subset C = ({x} T x ) of V (G[H]) is a restrained weakly connected independent dominating set of G[H] if and only if S is a weakly connected independent dominating set of G and T x is an independent dominating set of H for every x S. Proof : Suppose C is a restrained weakly connected independent dominating set of G[H]. By Theorem 3.1, C = ({x} T x ), where S is a weakly connected independent dominating set of G and T x is an independent dominating set of H for every x S. Conversely, suppose C = ({x} T x ), where S is a weakly connected independent dominating set of G and T x is an independent dominating set of H for every x S. By Theorem 3.1, C is a weakly connected independent dominating set of G[H]. Now, let (u, a) V (G[H])\C. Consider the following cases: Case 1. u S. Since S is a dominating set of G, choose w V (G)\S such that uw E(G). Hence, (w, a) V (G[H])\C and (u, a)(w, a) E(G[H]). Case 2. u / S. Since H is a nontrivial connected graph, there exists b V (H)\{a} such that ab E(H). Thus, (u, b) V (G[H])\C and (u, a)(u, b) E(G[H]). Therefore, C is a restrained weakly connected independent dominating set of G[H]. Corollary 3.3 Let G and H be nontrivial connected graphs. Then i rw (G[H]) = i w (G)i(H). Proof : Let C = ({x} T x ) be a minimum restrained weakly connected independent dominating set of G[H]. By Theorem 3.2, S is a weakly connected independent dominating set of G and T x is an independent dominating set of H for every x S. Hence, i rw (G[H]) = C = ({x} T x) = S Tx i w (G)i(H). Next, let S be a minimum weakly connected independent dominating set of G and D a minimum independent dominating set of H. For each x S, let T x = D. By Theorem 3.2, C = ({x} T x ) is a restrained weakly connected independent dominating set of G[H]. Thus,

6 978 Rene E. Leonida i rw (G[H]) C = ({x} T x) = S D = iw (G)i(H). Therefore, i rw (G[H]) = i w (G)i(H). References [1] S. R. Canoy, Jr., Restrained Domination in Graphs Under Some Binary Operations, Applied Mathematical Sciences, 8(2014), [2] G. S. Domke, J. H. Hattingh, S. T. Hedetniemi, R. C. Laskar, and L. R. Marcus, Restrained Domination in Graphs, Discrete Math., 203(1999), [3] R. E. Leonida, Weakly Connected Independent Dominations in the Join of Graphs, International Math. Forum, 8(2013), [4] R. E. Leonida and S. R. Canoy, Jr., Weakly Convex and Weakly Connected Independent Dominations in the Corona of Graphs, International Mathematical Forum, 8(2013), [5] R. E. Leonida, E. P. Sandueta, and S. R. Canoy, Jr., Weakly Connected Independent and Weakly Connected Total Dominations in a Product of Graphs, Applied Mathematical Sciences, 8(2014), [6] N. Tuan and S. R. Canoy, Jr. Independent Restrained Domination in Graphs, Applied Mathematical Sciences, 8(2014), Received: January 3, 2015; Published: February 1, 2015