The (n,m)-total domination number of a graph

Abstract

A set T of vertices in a graph G = (V,E) is an (n,m)-total dominating set of G if each vertex nu is an element of V is adjacent to at least n vertices in T and m vertices in V -T. The (n,m)-total domination number gamma(tn,m)(G) of G is the minimum cardinality of an (n,m)-total dominating set. In this communication, we characterize (n,m)-total dominating sets which are minimal, obtain some bounds on gamma(tn,m)(G) and its exact values for complete graphs, complete bipartite graphs, cycles and wheels are found. We obtain a sufficient condition on a gamma(tn)-set of G which is a gamma(tn,m)-set.