, Volume 28, Issue 2, pp 400-413

\(F_{3}\) -domination problem of graphs

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Abstract

Given a graph \(G\) and a set \(S\subseteq V(G),\) a vertex \(v\) is said to be \(F_{3}\) -dominated by a vertex \(w\) in \(S\) if either \(v=w,\) or \(v\notin S\) and there exists a vertex \(u\) in \(V(G)-S\) such that \(P:wuv\) is a path in \(G\) . A set \(S\subseteq V(G)\) is an \(F_{3}\) -dominating set of \(G\) if every vertex \(v\) is \(F_{3}\) -dominated by a vertex \(w\) in \(S.\) The \(F_{3}\) -domination number of \(G\) , denoted by \(\gamma _{F_{3}}(G)\) , is the minimum cardinality of an \(F_{3}\) -dominating set of \(G\) . In this paper, we study the \(F_{3}\) -domination of Cartesian product of graphs, and give formulas to compute the \(F_{3}\) -domination number of \(P_{m}\times P_{n}\) and \(P_{m}\times C_{n}\) for special \(m,n.\)