On Hop Roman Domination in Trees

Jafari Rad, Nader; Poureidi, Abolfazl

doi:10.22049/cco.2019.26469.1116

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	On Hop Roman Domination in Trees
Article 10, Volume 4, Issue 2, Summer and Autumn 2019, Page 201-208 PDF (436 K)
Document Type: Original paper
DOI: 10.22049/cco.2019.26469.1116
Authors
Nader Jafari Rad ¹; Abolfazl Poureidi²
¹Separtment of Mathemtics, Shahed University, Tehran, Iran
²Department of Mathematics, Shahrood University of Technology, Shahrood, Iran
Abstract
‎Let $G=(V,E)$ be a graph‎. ‎A subset $Ssubset V$ 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_{vin 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 $gamma_{hR}(T)=2gamma_{ch}(T)$ for a given‎ ‎tree $T$.\‎ ‎{bf Keywords:} hop dominating set‎, ‎connected hop dominating set‎, ‎hop Roman dominating‎ ‎function‎.
Keywords
hop dominating set; connected hop dominating set; hop Roman dominating function
Main Subjects
Graph theory


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