Complementary 3-Domination Number In Transformation Of Graphs

Abstract

In Graph theory, a dominating set of a graph is a subset  of its vertices such that every vertex in  is adjacent to atleast one vertex in. The minimum cardinality of a dominating set is called the domination number  A dominating set of a graph  is called a complementary 3-dominating set of  if for every vertex in  has atleast three neighbors in  The complementary 3-domination number  is the minimum cardinality taken over all complementary 3-dominating sets. The transformation graph of  is a simple graph with vertex set  in which adjacency is defined as follows Two elements in  are adjacent iff they are non adjacent in .  Two elements in  are adjacent iff they are non adjacent.  One element in  and one element in  are adjacent iff they are non adjacent in . It is denoted by  Let  be a simple undirected graph with order n and size m. The transformation graph of G is a simple graph with vertex set  in which adjacency is defined as follows:(a) Two elements in V(G) are adjacent iff they are adjacent in G. (b) Two elements in E(G) are adjacent iff they are non-adjacent in G. (c) One element in V(G) and one element in E(G) are adjacent iff they are incident in G.   It is denoted by . A transformation graph  of a simple graph G with vertex set  in which adjacency is defined by, (a) Two elements in  are adjacent iff they are non adjacet in .(b) Two elements in  are adjacent iff they are adjacent in.(c) One element in  and one element in  are adjacent iff they are non adjacent              in . In this paper we investigate some results in transformation graph  and obtain exact values for some graphs.