On a Conjecture for Power Domination

Abstract

The minimum cardinality of a power dominating set of a graph G is the power domination number of G, denoted by $\gamma _P(G)$. We prove a conjecture on power domination posed by Benson et al. (Discrete Appl Math 251:103–113, 2018), which states that if G is a graph on n vertices such that every component of G and its complement ${\overline{G}}$ have at least three vertices, then $\gamma _P(G)+\gamma _P({\overline{G}})\le \lfloor \frac{n}{3}\rfloor +2$. Also, we show that if G is a graph on n vertices such that both G and ${\overline{G}}$ are connected, then $\gamma _P(G)+\gamma _P({\overline{G}})\le \lceil \frac{n}{3}\rceil +1$. This result improves a previous result due to Bensen et al.