Exploring Algorithmic Solutions for the Independent Roman Domination Problem in Graphs

Given a graph $G=(V,E)$, a function $f:V\to {0,1,2}$ is said to be a \emph{Roman Dominating function} if for every $v\in V$ with $f(v)=0$, there exists a vertex $u\in N(v)$ such that $f(u)=2$. A Roman Dominating function $f$ is said to be an \emph{Independent Roman Dominating function} (or IRDF), if $V1\cup V2$ forms an independent set, where $Vi={v\in V~\vert~f(v)=i}$, for $i\in {0,1,2}$. The total weight of $f$ is equal to $\sum{v\in V} f(v)$, and is denoted as $w(f)$. The \emph{Independent Roman Domination Number} of $G$, denoted by $iR(G)$, is defined as min${w(f)~\vert~f$ is an IRDF of $G}$. For a given graph $G$, the problem of computing $iR(G)$ is defined as the \emph{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.