Note on Nordhaus-Gaddum problems for power domination

TL;DR

This paper improves bounds on the power domination number for graphs where both the graph and its complement are connected, providing tighter and sometimes tight bounds compared to previous domination number bounds.

Contribution

It introduces significantly improved upper bounds for the sum and product of the power domination number in specific graph classes, especially when both the graph and its complement are connected.

Findings

Tight upper sum bound for power domination number in connected graphs and their complements.

Improved upper bound for the product of power domination numbers under certain conditions.

Bound is significantly better than the known bounds for the domination number.

Abstract

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 must be connected. For these graphs, our bound is tight and is also significantly better than the corresponding bound for domination number. We also improve the product upper bound for the power domination number for graphs with certain properties.