International Journal of Applied Mathematics and Computation Journal homepage: www.darbose.in/ijamc ISSN: 0974 - 4665 (Print) 0974 - 4673 (Online) Volume 4(4) 2012 352–358 Restrained 2-Domination Number of Complete Grid Graphs J. Jagan Mohan a , Indrani Kelkar b a Fluid Dynamics Division, School of Advanced Sciences, VIT University, Vellore, Tamil Nadu - 632014, India. b Department of Mathematics, Acharya Institute of Technology, Bangalore, India. ABSTRACT In this paper we present 2-dominating sets for complete grid graphs G k,n by using V-merge and H-merge techniques recursively and then find 2-domination and restrained 2-domination numbers of complete grid graphs. Further we verify the Vizings conjecture for both domination parameters. Keywords: Complete grid graph; cartesian product of graphs; merging of graphs; domination number; 2-domination number. c  2012 Darbose. All rights reserved. 1. Introduction Starting in the eighties, domination number of cartesian products were intensively investigated. In the meantime, some papers on domination number of cardinal products of graphs were also published. The study of domination number of product of graphs was initiated by Vizing [8]. He conjectured that the domination number of the cartesian product of two graphs is always greater than or equal to the product of the domination numbers of the two factors. This conjecture is still unproven. For complete grid graphs P k × P n , algorithms were given for a fixed k which compute γ (P k × P n ) in O(n) time. In fact, the domination number problem for k × n grids, where k is fixed, has a constant time solution. Other domination parameters like total domination number and double domination number have been found for specific value of k. In this paper we find the 2-domination and restrained 2-domination numbers of complete grid graph G k,n = P k × P n . 2. Notations and Terminology For notations and graph theory terminology, we follow [1]. Let G =(V,E) be a simple graph with vertex set V and edge set E. A subset D of V in a graph G is said to be a dominating set of G if every vertex of V \ D is adjacent to at least one vertex in D. The domination number γ (G) is the minimum cardinality of a dominating set of G. The concept of 2-domination was introduced by Fink and Jacobson [3]. A dominating set D ⊆ V (G) is a 2- dominating set if every vertex of V \ D is adjacent to at least two vertices in D. The 2-domination number of G denoted by γ 2 (G) is the minimum cardinality of a 2-dominating set of G. The concept of restrained 2- domination was introduced by Kelkar and Mohan [5]. A set D ⊆ V (G) is a restrained 2-dominating set of G if every vertex of V \ D is adjacent to at least two vertices in D and every vertex of V \ D is adjacent to a vertex in V \ D. The restrained 2-domination number of G denoted by γ r2 (G) is the minimum cardinality of a restrained 2-dominating set of G. Email address: j.jaganmohan@hotmail.com (J. Jagan Mohan) 352