Algorithmic complexity of outer independent Roman domination and outer independent total Roman domination

Abstract

Let $$G=(V,E)$$ be a graph. A function $$f : V \rightarrow \{0, 1, 2\}$$ is an outer independent Roman dominating function (OIRDF) on a graph G if for every vertex $$v \in V$$ with $$f (v) = 0$$ there is a vertex u adjacent to v with $$f (u) = 2$$ and $$\{x\in V:f(x)=0\}$$ is an independent set. The weight of f is the value $$ f(V)=\sum _{v\in V}f(v)$$ . An outer independent total Roman dominating function (OITRDF) f on G is an OIRDF on G such that for every $$v\in V$$ with $$f(v)>0$$ there is a vertex u adjacent to v with $$f (u)>0$$ . The minimum weight of an OIRDF on G is called the outer independent Roman domination number of G, denoted by $$\gamma _{oiR}(G)$$ . Similarly, the outer independent total Roman domination number of G is defined, denoted by $$\gamma _{oitR}(G)$$ . In this paper, we first show that computing $$\gamma _{oiR}(G)$$ (respectively, $$\gamma _{oitR}(G)$$ ) is a NP-hard problem, even when G is a chordal graph. Then, for a given proper interval graph $$G=(V,E)$$ we propose an algorithm to compute $$\gamma _{oiR}(G)$$ (respectively, $$\gamma _{oitR}(G)$$ ) in $${\mathcal {O}}(|V| )$$ time.