On the independence transversal total domination number of graphs

A total dominating set of a graph G having non empty intersection with all the independent sets of maximum cardinality in G is an independent transversal total dominating set. The minimum cardinality of any independent transversal total dominating set is denoted by γtt(G). In this paper we introduce this concept and begin the study of its mathematical properties. Specifically, we prove that the complexity of the decision problem associated to the computation of the value of γtt(G) is NP-complete, under the assumption that the independence number is known. Moreover, we present tight lower and upper bounds on γtt(G) and give some realizability results in concordance with these bounds. For instance, we show that for any two positive integers a,b such that 2≤a≤2b3 there is a graph G of order b such that γtt(G)=a.