# On Hop Roman Domination in Trees

Document Type : Original paper

Authors

1 Separtment of Mathemtics, Shahed University, Tehran, Iran

2 Department of Mathematics, Shahrood University of Technology, Shahrood, Iran

Abstract

‎Let $G=(V,E)$ be a graph. A subset $S\subset V$ is a hop dominating set if every vertex outside $S$ is at distance two from a vertex of $S$. A hop dominating set $S$ which induces a connected subgraph is called a connected hop dominating set of $G$. The connected hop domination number of $G$, $\gamma_{ch}(G)$, is the minimum cardinality of a connected hop dominating set of $G$. A hop Roman dominating function (HRDF) of a graph $G$ is a function $f: V(G)\longrightarrow \{0, 1, 2\}$ having the property that for every vertex $v \in V$ with $f(v) = 0$ there is a vertex $u$ with $f(u)=2$ and $d(u,v)=2$. The weight of an HRDF $f$ is the sum $f(V) = \sum_{v\in V} f(v)$. The minimum weight of an HRDF on $G$ is called the hop Roman domination number of $G$ and is denoted by $\gamma_{hR}(G)$. We give an algorithm that decides whether $\gamma_{hR}(T)=2\gamma_{ch}(T)$ for a given tree $T$.

Keywords

Main Subjects

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