﻿ On Hop Roman Domination in Trees

# On Hop Roman Domination in Trees

##### نویسندگان
• Abolfazl Poureidi Department of Mathematics, Shahrood University of Technology, Shahrood, Iran
##### چکیده

‎Let \$G=(V,E)\$ be a graph‎. ‎A subset \$Ssubset V\$ is a hop dominating set‎‎if 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\$‎. ‎The‎‎connected hop domination number of \$G\$‎, ‎\$ gamma_{ch}(G)\$,‎‎‎ ‎is the minimum cardinality of a connected hop‎‎dominating set of \$G\$‎. ‎A hop‎‎Roman dominating function (HRDF) of a graph \$G\$ is a function \$‎‎f‎: ‎V(G)longrightarrow {0‎, ‎1‎, ‎2} \$ having the property that‎‎for every vertex \$ v in V \$ with \$ f(v) = 0 \$ there is a‎‎vertex \$ u \$ with \$ f(u)=2 \$ and \$ d(u,v)=2 \$‎.‎The weight of‎‎an HRDF \$ f \$ is the sum \$f(V) = sum_{vin V} f(v) \$‎. ‎The‎‎minimum weight of an HRDF on \$ G \$ is called the hop Roman‎‎domination number of \$ G \$ and is denoted by \$ gamma_{hR}(G)‎‎\$‎. ‎We give an algorithm‎‎that decides whether \$gamma_{hR}(T)=2gamma_{ch}(T)\$ for a given‎‎tree \$T\$.‎‎{bf Keywords:} hop dominating set‎, ‎connected hop dominating set‎, ‎hop Roman dominating‎‎function‎.

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عنوان ژورنال:

دوره 4  شماره 2

صفحات  201- 208

تاریخ انتشار 2019-12-01

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