Power domination in cubic graphs and Cartesian products

The power domination problem focuses on finding the optimal placement of phase measurement units (PMUs) to monitor an electrical power network. In the context of graphs, the power domination number of a graph $G$, denoted $\gamma_P(G)$, is the minimum number of vertices needed to observe every vertex in the graph according to a specific set of observation rules. In \cite{ZKC_cubic}, Zhao et al. proved that if $G$ is a connected claw-free cubic graph of order $n$, then $\gamma_P(G) \leq n/4$. In this paper, we show that if $G$ is a claw-free diamond-free cubic graph of order $n$, then $\gamma_P(G) \le n/6$, and this bound is sharp. We also provide new bounds on $\gamma_P(G \Box H)$ where $G\Box H$ is the Cartesian product of graphs $G$ and $H$. In the specific case that $G$ and $H$ are trees whose power domination number and domination number are equal, we show the Vizing-like inequality holds and $\gamma_P(G \Box H) \ge \gamma_P(G)\gamma_P(H)$.