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.
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We wish to thank the anonymous reviewers for valuable comments, which helped us improve the quality of the presentation of the paper. This work was supported by the National Science Council of Taiwan under Grants NSC100-2221-E-007-108-MY3, NSC102-2221-E-007-075-MY3 and NSC100-3113-P-002-012
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Liao, CS. Power domination with bounded time constraints. J Comb Optim 31, 725–742 (2016). https://doi.org/10.1007/s10878-014-9785-2
- Power domination
- Time constraint