Power domination with bounded time constraints

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 $$S$$ S is a power dominating set (PDS) of a graph $$G=(V,E)$$ G = ( V , E ) if every vertex $$v$$ v in $$V$$ V can be observed under the following two observation rules: (1) $$v$$ v is dominated by $$S$$ S , i.e., $$v \in S$$ v ∈ S or $$v$$ v has a neighbor in $$S$$ S ; and (2) one of $$v$$ v ’s neighbors, say $$u$$ u , and all of $$u$$ u ’s neighbors, except $$v$$ 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.