The total global domination number of a graph

Abstract

A total dominating set T of a graph G = (V, E) is a total global dominating set (t.g.d. set) if T is also a total dominating set of ?. The total global domination number ?tg (G) of G is the minimum cardinality of a t.g.d. set. In this paper, we characterize t.g.d. sets and bounds are obtained for ?tg (G). We exhibit inequalities involving variations on domination numbers and vertex covering number. For graphs with diameter at least five, three of the four domination numbers considered turn out to be identical. Values are given for paths, cycles and complete bipartite graphs. We characterize graphs with | V | vertices in any minimum total global dominating set.