Failed power domination on graphs

• Published in: arXiv.org
• Main Authors: Glasser, Abraham, Jacob, Bonnie, Lederman, Emily, Radziszowski, Stanisław
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 04.09.2019
• Subjects: Apexes; Graphs; Neighborhoods
• ISSN: 2331-8422

Let \(G\) be a simple graph with vertex set \(V\) and edge set \(E\), and let \(S \subseteq V\). The \emph{open neighborhood} of \(v \in V\), \(N(v)\), is the set of vertices adjacent to \(v\); the \emph{closed neighborhood} is given by \(N[v] = N(v) \cup \{v\}\). The \emph{open neighborhood} of \(S\), \(N(S)\), is the union of the open neighborhoods of vertices in \(S\), and the \emph{closed neighborhood} of \(S\) is \(N[S] = S \cup N(S)\). The sets \( \mathcal{P}^i(S), i \geq 0\), of vertices \emph{monitored} by \(S\) at the \(i^{\ {th}}\) step are given by \(\mathcal{P}^0(S) = N[S]\) and \(\mathcal{P}^{i+1}(S) = \mathcal{P}^i(S) \bigcup\left\{ w : \{ w \} = N[v] \backslash \mathcal{P}^i(S) \ { for some } v \in \mathcal{P}^i(S) \right\}\). If there exists \(j\) such that \(\mathcal{P}^j(S) = V\), then \(S\) is called a \emph{power dominating set}, PDS, of \(G\). We introduce and discuss the \emph{failed power domination number} of a graph \(G\), \(\bar{\gamma}_p(G)\), the largest cardinality of a set that is not a PDS. We prove that \(\bar{\gamma}_p(G)\) is NP-hard to compute, determine graphs in which every vertex is a PDS, and compare \(\bar{\gamma}_p(G)\) to similar parameters.