Nordhaus–Gaddum problems for power domination

A power dominating set of a graph G is a set S of vertices that can observe the entire graph under the rules that (1) the closed neighborhood of every vertex in S is observed, and (2) if a vertex and all but one of its neighbors are observed, then the remaining neighbor is observed; the second rule is applied iteratively. The power domination number of G, denoted by γP(G), is the minimum number of vertices in a power dominating set. A Nordhaus–Gaddum problem for power domination is to determine a tight lower or upper bound on γP(G)+γP(G¯) or γP(G)⋅γP(G¯), where G¯ denotes the complement of G. The upper and lower Nordhaus–Gaddum bounds over all graphs for the power domination number follow from known bounds on the domination number and examples. In this note we improve the upper sum bound for the power domination number substantially for graphs having the property that both the graph and its complement are connected. For these graphs, our bound is tight and is also significantly better than the corresponding bound for the domination number. We also improve the product upper bound for the power domination number for graphs with certain properties.