  Power domination in graphs arises from the problem of monitoring an electric
power system by placing as few measurement devices in the system as possible. A
power dominating set of a graph is a set of vertices that observes every vertex
in the graph, following a set of rules for power system monitoring. A practical
problem of interest is to determine the minimum number of additional
measurement devices needed to monitor a power network when the network is
expanded and the existing devices remain in place. In this paper, we study the
problem of finding the smallest power dominating set that contains a given set
of vertices X. We also study the related problem of finding the smallest zero
forcing set that contains a given set of vertices X. The sizes of such sets in
a graph G are respectively called the restricted power domination number and
restricted zero forcing number of G subject to X. We derive several tight
bounds on the restricted power domination and zero forcing numbers of graphs,
and relate them to other graph parameters. We also present exact and
algorithmic results for computing the restricted power domination number,
including integer programs for general graphs and a linear time algorithm for
graphs with bounded treewidth. We also use restricted power domination to
obtain a parallel algorithm for finding minimum power dominating sets in trees.
