Outer-independent total Roman domination in graphs

Given a graph G with vertex set V, a function f : V -> {0, 1, 2) is an outer-independent total Roman dominating function on G if every vertex v is an element of V for which f(u) = 0 is adjacent to at least one vertex u is an element of V such that f (u) = 2, every vertex x is an element of V for which f (x) >= 1 is adjacent to at least one vertex y is an element of V such that f (y) >= 1, and any two different vertices a, b for which f (a) = f (b) = 0 are not adjacent. The minimum weight omega(f) = Sigma(w is an element of V) f(w) of any outer-independent total Roman dominating function on G is the outer-independent total Roman domination number, gamma(oitR)(G), of G. In this article, we introduce the concepts above and begin with the study of its combinatorial and computational properties. For instance, we give several closed relationships between this parameter and other ones related with domination in graphs. We prove that computing gamma(oitR) of a graph G is an NP-hard problem. In addition, we present some closed formulas for gamma(oitR)(G) in the cases G represents some special families of graphs. (C) 2018 Elsevier B.V. All rights reserved.