IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 _______________________________________________________________________________________ Volume: 02 Issue: 09 | Sep-2013, Available @ http://www.ijret.org 454 BOUNDS ON DOUBLE DOMINATION IN SQUARES OF GRAPHS M. H. Muddebihal 1 , Srinivasa G 2 1 Professor, Department of Mathematics, Gulbarga University, Karnataka, India, mhmuddebihal@yahoo.co.in 2 Assistant Professor, Department of Mathematics, B. N. M. I. T , Karnataka, India, gsgraphtheory@yahoo.com Abstract Let the square of a graph G , denoted by 2 G has same vertex set as in G and every two vertices u and v are joined in 2 G if and only if they are joined in G by a path of length one or two. A subset D of vertices of 2 G is a double dominating set if every vertex in 2 G is dominated by at least two vertices of D . The minimum cardinality double dominating set of 2 G is the double domination number, and is denoted by ( 29 2 d G γ . In this paper, many bounds on ( 29 2 d G γ were obtained in terms of elements of G . Also their relationship with other domination parameters were obtained. Key words: Graph, Square graph, Double dominating set, Double domination number. Subject Classification Number: AMS-05C69, 05C70. --------------------------------------------------------------------***---------------------------------------------------------------------- 1. INTRODUCTION In this paper, we follow the notations of [1]. All the graphs considered here are simple, finite and connected. As usual ( ) p VG = and ( ) q EG = denote the number of vertices and edges of G , respectively. In general, we use X 〈 〉 to denote the subgraph induced by the set of vertices X and () Nv and [ ] Nv denote the open and closed neighborhoods of a vertex v , respectively. The notation ( 29 ( 29 ( 29 0 1 G G α α is the minimum number of vertices(edges) is a vertex(edge) cover of G . Also ( 29 ( 29 ( 29 0 1 G G β β is the minimum number of vertices (edges) is a maximal independent set of vertex (edge) of G . Let ( 29 deg v is the degree of a vertex v and as usual ( 29 ( 29 ( 29 G G δ ∆ denote the minimum (maximum) degree of G . A vertex of degree one is called an end vertex and its neighbor is called a support vertex. Suppose a support vertex v is adjacent to at least two end vertices then it is called a strong support vertex. A vertex v is called cut vertex if removing it from G increases the number of components of G . The distance between two vertices u and v is the length of the shortest uv - path in G . The maximum distance between any two vertices in G is called the diameter, denoted by ( 29 diam G . The square of a graph G , denoted by 2 G has the same vertex set as in G and the two vertices u and v are joined in 2 G if and only if they are joined in G by a path of length one or two (see [1], [2]). We begin by recalling some standard definitions from domination theory. A set S V ⊆ is said to be a double dominating set of G , if every vertex of G is dominated by at least two vertices of S . The double domination number of G is denoted by ( ) d G γ and is the minimum cardinality of a double dominating set of G . This concept was introduced by F. Harary and T. W. Haynes [3]. A dominating set ( ) S VG ⊆ is a restrained dominating set of G , if every vertex not in S is adjacent to a vertex in S and to a vertex in V S - . The restrained domination number of G , denoted by ( 29 re G γ is the minimum cardinality of a restrained dominating set of G . This concept was introduced by G. S. Domke et. al.,[4]. A dominating set ( ) S VG ⊆ is said to be connected dominating set of G , if the subgraph S is not disconnected. The minimum cardinality of vertices in such a set is called the connected domination number of G and is denoted by ( ) c G γ [5]. A subset ( 29 2 D VG ⊆ is said to be a dominating set of 2 G , if every vertex not in D is adjacent to some vertex in D . The domination number of 2 G , denoted by ( 2 G γ , is the