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.