Connected power domination in graphs

TL;DR

This paper investigates the problem of finding the smallest connected set of vertices in a graph that can monitor the entire network, providing complexity results, structural insights, and optimization formulations.

Contribution

It introduces the concept of connected power domination, proves NP-hardness, offers efficient algorithms for special graph classes, and develops integer programming models.

Findings

Connected power domination number is NP-hard in general.

Linear-time algorithms for cactus and block graphs.

New integer programming formulations and computational results.

Abstract

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.