Non-Split Total Domination Number and Inverse Non-Split Total Domination Number of Some Special Graphs

Abstract

A set D is said to be a dominating set if every vertex  is adjacent to some vertex in D. The minimum cardinality of the dominating set is the domination number denoted as   A dominating set D is said to be a non-split dominating set if  is connected. The non-split domination number is denoted by

A set  of vertices in a graph G is called the total dominating set if every vertex   is adjacent to an element of . The minimum cardinality of the total dominating set is the total domination number is denoted by

In this paper a new parameter, Non-split total dominating set, the non-split total domination number  and also the inverse non-split total domination number have been introduced. A total dominating set  is said to be a non-split total dominating set if  is connected and every vertex  is adjacent to an element of . The minimum non-split total domination number is denoted by  Further a total dominating set . If the complement set  contains its own total ′ and the induced subgraph is connected, then ′ is an inverse non-split total dominating set. The minimum cardinality of such a set is the inverse non-split total domination number, denoted as  In this paper we have obtained Non-split total domination number and an Inverse Non-split total domination number of some special graphs.