Further results on the independent roman domination number of graphs

Abstract

Let f : V (G) → {0, 1, 2} be a function on a graph G with vertex set V (G). Let Vi = {v ∈ V (G) : f(v) = i} for every i ∈ {0, 1, 2}. The function f is said to be an independent Roman dominating function on G if V₁ ∪ V₂ is an independent set and N(v) ∩ V₂ ̸= ∅ for every v ∈ V0. The minimum weight ω(f) = ∑ x∈V (G) f(x) among all independent Roman dominating functions f on G is the independent Roman domination number of G, and is denoted by iR(G). In this  paper we continue with the study of this parameter. In particular, we provide new bounds on iR(G) in terms of other domination invariants. Some of our  results are tight bounds that improve some well-known results. Finally, we compute the independent Roman domination number of some product  graphs, and we provide an alternative proof to show that the problem of computing iR(G) is NP-hard.