Roman domination number of Generalized Petersen Graphs P(n,2)

A \(Roman\ domination\ function\) on a graph \(G=(V, E)\) is a function \(f:V(G)\rightarrow\{0,1,2\}\) satisfying the condition that every vertex \(u\) with \(f(u)=0\) is adjacent to at least one vertex \(v\) with \(f(v)=2\). The \(weight\) of a Roman domination function \(f\) is the value \(f(V(G))...

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Bibliographic Details
Published inarXiv.org
Main Authors Wang, Haoli, Xu, Xirong, Yang, Yuansheng, Ji, Chunnian
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 12.03.2011
Subjects
Graphs
Mathematical functions
Minimum weight
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Summary:A \(Roman\ domination\ function\) on a graph \(G=(V, E)\) is a function \(f:V(G)\rightarrow\{0,1,2\}\) satisfying the condition that every vertex \(u\) with \(f(u)=0\) is adjacent to at least one vertex \(v\) with \(f(v)=2\). The \(weight\) of a Roman domination function \(f\) is the value \(f(V(G))=\sum_{u\in V(G)}f(u)\). The minimum weight of a Roman dominating function on a graph \(G\) is called the \(Roman\ domination\ number\) of \(G\), denoted by \(\gamma_{R}(G)\). In this paper, we study the {\it Roman domination number} of generalized Petersen graphs P(n,2) and prove that \(\gamma_R(P(n,2)) = \lceil {\frac{8n}{7}}\rceil (n \geq 5)\).
ISSN:2331-8422