Exploring algorithmic solutions for the Independent Roman Domination problem in graphs

Abstract

Given a graph $G=(V,E)$, a function $f:V\to \{0,1,2\}$ is said to be a Roman Dominating function if for every $v\in V$ with $f\left(v\right)=0$, there exists a vertex $u\in N\left(v\right)$ such that $f\left(u\right)=2$. A Roman Dominating function $f$ is said to be an Independent Roman Dominating function (or IRDF), if $V_1\cup V_2$ forms an independent set, where $V_i=\{v\in V|f\left(v\right)=i\}$, for $i\in \{0,1,2\}$. The total weight of $f$ is equal to ${\sum }_{v\in V}f\left(v\right)$, and is denoted as $w\left(f\right)$. The Independent Roman Domination Number of $G$, denoted by $i_R\left(G\right)$, is defined as min{$w\left(f\right)|f$ is an IRDF of $G$}. For a given graph $G$, the problem of computing $i_R\left(G\right)$ is defined as the Minimum Independent Roman Domination problem. The problem is already known to be NP-hard for bipartite graphs. In this paper, we further study the algorithmic complexity of the problem. In this paper, we propose a polynomial-time algorithm to solve the Minimum Independent Roman Domination problem for distance-hereditary graphs, split graphs, and $P_4$-sparse graphs.