  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.
