  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.

Combinatorics