On the \(p\)-reinforcement and the complexity

Abstract

Let \(G=(V,E)\) be a graph and \(p\) be a positive integer. A subset \(S\subseteq V\) is called a \(p\)-dominating set if each vertex not in \(S\) has at least \(p\) neighbors in \(S\). The \(p\)-domination number \(\gamma _p(G)\) is the size of a smallest \(p\)-dominating set of \(G\). The \(p\)-reinforcement number \(r_p(G)\) is the smallest number of edges whose addition to \(G\) results in a graph \(G^{\prime }\) with \(\gamma _p(G^{\prime })< \gamma _p(G)\). In this paper, we give an original study on the \(p\)-reinforcement, determine \(r_p(G)\) for some graphs such as paths, cycles and complete \(t\)-partite graphs, and establish some upper bounds on \(r_p(G)\). In particular, we show that the decision problem on \(r_p(G)\) is NP-hard for a general graph \(G\) and a fixed integer \(p\ge 2\).