Equality of total domination and chromatic total domination in graphs

Abstract

Let be a simple, finite and undirected graph and without isolated vertex. A subset D of V is said to be dominating set if for every in there exist a vertex in such that and are adjacent. The minimum cardinality of a dominating set of is called the domination number of and is denoted by . is minimal dominating set of a graph if no proper subset of is a dominating set of . is a total dominating set of if has no isolates. The minimum cardinality of a total dominating set of is called the total domination number of and is denoted by . A Subset D of V is said to be a chromatic total dominating set if D is a total dominating set and =. The minimum cardinality of the chromatic total dominating set is called a chromatic total dominating number .In any graph G ,every chromatic total dominating set is a total dominating set.But converse is not true.In some graphs ,every set is a chromatic total dominating set.