  The study of power domination in graphs arises from the problem of placing a
minimum number of measurement devices in an electrical network while monitoring
the entire network. A power dominating set of a graph is a set of vertices from
which every vertex in the graph can be observed, following a set of rules for
power system monitoring. In this paper, we study the problem of finding a
minimum power dominating set which is connected; the cardinality of such a set
is called the connected power domination number of the graph. We show that the
connected power domination number of a graph is NP-hard to compute in general,
but can be computed in linear time in cactus graphs and block graphs. We also
give various structural results about connected power domination, including a
cut vertex decomposition and a characterization of the effects of various
vertex and edge operations on the connected power domination number. Finally,
we present novel integer programming formulations for power domination,
connected power domination, and power propagation time, and give computational
results.
