On Hop Roman Domination in Trees

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

• hop dominating set
• connected hop dominating set
• hop Roman dominating function