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.