  A power dominating set of a graph is a set of vertices that observes every
vertex in the graph by combining classical domination with an iterative
propagation process arising from electrical circuit theory. In this paper, we
study the power domination polynomial of a graph $G$ of order $n$, defined as
$\mathcal{P}(G;x)=\sum_{i=1}^n p(G;i) x^i$, where $p(G;i)$ is the number of
power dominating sets of $G$ of size $i$. We relate the power domination
polynomial to other graph polynomials, present structural and extremal results
about its roots and coefficients, and identify some graph parameters it
contains. We also derive decomposition formulas for the power domination
polynomial, compute it explicitly for several families of graphs, and explore
graphs which can be uniquely identified by their power domination polynomials.
