Power domination with bounded time constraints

Abstract

Based on the power observation rules, the problem of monitoring a power utility network can be transformed into the graph-theoretic power domination problem, which is an extension of the well-known domination problem. A set $S$ is a power dominating set (PDS) of a graph $G=(V,E)$ if every vertex $v$ in $V$ can be observed under the following two observation rules: (1) $v$ is dominated by $S$, i.e., $v \in S$ or $v$ has a neighbor in $S$; and (2) one of $v$’s neighbors, say $u$, and all of $u$’s neighbors, except $v$, can be observed. The power domination problem involves finding a PDS with the minimum cardinality in a graph. Similar to message passing protocols, a PDS can be considered as a dominating set with propagation that applies the second rule iteratively. This study investigates a generalized power domination problem, which limits the number of propagation iterations to a given positive integer; that is, the second rule is applied synchronously with a bounded time constraint. To solve the problem in block graphs, we propose a linear time algorithm that uses a labeling approach. In addition, based on the concept of time constraints, we provide the first nontrivial lower bound for the power domination problem.