From Total Roman Domination in Lexicographic Product Graphs to Strongly Total Roman Domination in Graphs

Let G be a graph with no isolated vertex and let N (v) be the open neighbourhood of v is an element of V (G). Let f : V (G) -> {0, 1, 2} be a function and V-i = {v is an element of V (G) : f (v) = i} for every i is an element of{0, 1, 2}. We say that f is a strongly total Roman dominating function on G if the subgraph induced by V-1 boolean OR V-2 has no isolated vertex and N (v) boolean AND V-2 not equal empty set for every v is an element of V (G) \ V2. The strongly total Roman domination number of G, denoted by gamma(s)(tR) (G), is defined as the minimum weight omega(f) = Sigma(x is an element of V(G)) f (x) among all strongly total Roman dominating functions f on G. This paper is devoted to the study of the strongly total Roman domination number of a graph and it is a contribution to the Special Issue "Theoretical Computer Science and Discrete Mathematics" of Symmetry. In particular, we show that the theory of strongly total Roman domination is an appropriate framework for investigating the total Roman domination number of lexicographic product graphs. We also obtain tight bounds on this parameter and provide closed formulas for some product graphs. Finally and as a consequence of the study, we prove that the problem of computing gamma(s)(tR) (G) is NP-hard.