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) 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.