Total Roman {2}-domination in graphs

Abstract

Given a graph G = (V, E), a function f: V → {0, 1, 2} is a total Roman {2}-dominating function if

every vertex ʋ ∈ V for which f (ʋ) = 0 satisfies that ∑ _{u ∈N (ʋ)} f (ʋ) ≥ 2, where N (ʋ) represents the open neighborhood of ʋ, and

every vertex x ∈ V for which f (x) ≥ 1 is adjacent to at least one vertex y ∈ V such that f (y) ≥ 1.

The weight of the function f is defined as ω(f ) = ∑ _{ʋ ∈V} f (ʋ). The total Roman {2}-domination number, denoted by γ_{t {R2}}(G), is the minimum weight among all total Roman {2}-dominating functions on G. In this article we introduce the concepts above and begin the study of its combinatorial and computational properties. For instance, we give several closed relationships between this parameter and other domination related parameters in graphs. In addition, we prove that the complexity of computing the value γ_{t {R2}}(G) is NP-hard, even when restricted to bipartite or chordal graphs.