On a Relationship between \(2\)-dominating and \(5\)-dominating sets in Graphs

In a graph, a set \(D\) is an \(n\)-dominating set if for every vertex \(x\), not in \(D\), \(x\) is adjacent to at least \(n\) vertices of \(D\). The \(n\)-domination number, \(\gamma_n(G)\), is the order of a smallest \(n\)-dominating set. When this concept was first introduced by Fink and Jacobson, they asked whether there existed a function \(f(n)\), such that if \(G\) is any graph with minimum degree at least \(n\), then \(\gamma_n(G) < \gamma_{f(n)}(G)\). In this paper we show that \(\gamma_2(G) < \gamma_5(G)\) for all graphs with minimum degree at least \(2\). Further, this result is best possible in the sense that there exist infinitely many graphs \(G\) with minimum degree at least \(2\) having \(\gamma_2(G) = \gamma_4(G)\).