论文标题
满足笛卡尔产品的功率统治的巨大限制的图形
Graphs which satisfy a Vizing-like bound for power domination of Cartesian products
论文作者
论文摘要
功率支配是一个两步观察过程，用于监视电源网络，可以看作是统治和零强迫的组合。给定图形$ g $，可以使用此过程观察$ g $的所有顶点的子集$ s \ subseteq v（g）$，被称为$ g $的电力支配集，而功率支配数量为$ g $，$γ_p（g）$，是电动机中的最小角度数量。我们在图的顶点上介绍了一个新分区，以为功率支配号码提供下限。我们还考虑了两个图的笛卡尔产品的功率支配数字，即$ g \ box h $，并显示某些图满足了与功率支配数字有关的杂色型。特别是，我们证明，对于任何两棵树，$ t_1 $和$ t_2 $，$γ_p（t_1）γ_p（t_2）\ leqγ_p（t_1 \ box t_2）$。
Power domination is a two-step observation process that is used to monitor power networks and can be viewed as a combination of domination and zero forcing. Given a graph $G$, a subset $S\subseteq V(G)$ that can observe all vertices of $G$ using this process is known as a power dominating set of $G$, and the power domination number of $G$, $γ_P(G)$, is the minimum number of vertices in a power dominating set. We introduce a new partition on the vertices of a graph to provide a lower bound for the power domination number. We also consider the power domination number of the Cartesian product of two graphs, $G \Box H$, and show certain graphs satisfy a Vizing-like bound with regards to the power domination number. In particular, we prove that for any two trees $T_1$ and $T_2$, $γ_P(T_1)γ_P(T_2) \leq γ_P(T_1 \Box T_2)$.
