# Efficient algorithms for independent Roman domination on some classes of graphs

Document Type : Original paper

Author

Shahrood University of Technology

Abstract

Let $G=(V,E)$ be a given graph of order $n$. A function $f : V \to \{0,1, 2\}$ is an independent Roman dominating function (IRDF) on $G$ if for every vertex $v\in V$ with $f(v)=0$ there is a vertex $u$ adjacent to $v$ with $f(u)=2$ and $\{v\in V:f(v)> 0\}$ is an independent set. The weight of an IRDF $f$ on $G$ is the value $f(V)=\sum_{v\in V}f(v)$. The minimum weight of an IRDF among all IRDFs on $G$ is called the independent Roman domination number of~$G$. In this paper, we give algorithms for computing the independent Roman domination number of $G$ in $O(|V|)$ time when $G=(V,E)$ is a tree, unicyclic graph or proper interval graph.

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