论文标题
立方图和笛卡尔产品中的功率支配
Power domination in cubic graphs and Cartesian products
论文作者
论文摘要
功率支配问题的重点是找到相位测量单元（PMU）的最佳放置以监视电力网络。在图形的上下文中，图$ g $的功率支配数，表示为$γ_p（g）$，是根据特定的观察规则集观察图表中每个顶点所需的最小顶点。在\ cite {zkc_cubic}，Zhao等。证明，如果$ g $是连接的订单$ n $的无爪立方图，则$γ_p（g）\ leq n/4 $。在本文中，我们表明，如果$ g $是无爪的无钻石立方图$ n $，则是$γ_p（g）\ le n/6 $，并且这种界限很敏锐。我们还提供了$γ_p（g \ box h）$的新界限，其中$ g \ box h $是图形$ g $和$ h $的笛卡尔产品。在特定情况下，$ g $和$ h $是树木的功率支配数字和统治数量相等的树，我们表明类似杂色的不平等含量和$γ_p（g \ box h）\geγ_p（g）γ_p（h）$。
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 $γ_P(G)$, 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 $γ_P(G) \leq n/4$. In this paper, we show that if $G$ is a claw-free diamond-free cubic graph of order $n$, then $γ_P(G) \le n/6$, and this bound is sharp. We also provide new bounds on $γ_P(G \Box H)$ 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 $γ_P(G \Box H) \ge γ_P(G)γ_P(H)$.
