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 $V_1\cup V_2$ forms an independent set, where $V_i=\{v\in V~\vert~f(v)=i\}$, 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 $i_R(G)$, is defined as min$\{w(f)~\vert~f$ is an IRDF of $G\}$. For a given graph $G$, the problem of computing $i_R(G)$ 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.

翻译：给定图$G=(V,E)$，若函数$f:V\to \{0,1,2\}$满足：对每个满足$f(v)=0$的顶点$v\in V$，均存在相邻顶点$u\in N(v)$使得$f(u)=2$，则称$f$为\emph{罗马控制函数}。若罗马控制函数$f$还满足$V_1\cup V_2$构成独立集（其中$V_i=\{v\in V~\vert~f(v)=i\}$，$i\in \{0,1,2\}$），则称$f$为\emph{独立罗马控制函数}（简称IRDF）。函数$f$的总权重定义为$w(f)=\sum_{v\in V} f(v)$。图$G$的\emph{独立罗马控制数}记作$i_R(G)$，定义为$i_R(G)=$min$\{w(f)~\vert~f$是$G$的IRDF$\}$。对于给定图$G$，计算$i_R(G)$的问题称为\emph{最小独立罗马控制问题}。已知该问题对于二分图是NP难问题。本文进一步研究该问题的算法复杂性，提出了多项式时间算法求解距离遗传图、分裂图和$P_4$-稀疏图上的最小独立罗马控制问题。