Failed power domination on graphs

Abstract

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.