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.

功率控制集的曲线图中的最小基数ģ是的功率控制数ģ，记\（\伽马_P（G）\） 。我们证明了本森（Benson）等人对权力支配的猜想。（离散应用数学251：103-113，2018年），其中指出，如果G是n个顶点上的图，使得G的每个分量及其补数\（{\ overline {G}} \）至少具有三个顶点，然后\（\ gamma _P（G）+ \ gamma _P（{\ overline {G}}）\ le \ lfloor \ frac {n} {3} \ rfloor +2 \）。同样，我们表明如果G是n个顶点上的图，则G和\（{\ overline {G}} \）已连接，然后\（\ gamma _P（G）+ \ gamma _P（{\ overline {G}}）\ le \ lceil \ frac {n} {3} \ rceil +1 \）。由于本森等人，该结果改善了先前的结果。