In this paper, we first show that the power domination number of a connected $4$-regular claw-free graph on $n$ vertices is at most $\frac{n+1}{5}$, and the bound is sharp. The statement partly disprove the conjecture presented by Dorbec et al. in SIAM J. Discrete Math., 27:1559-1574, 2013. Then we present a dynamic programming style linear-time algorithm for weighted power domination problem in trees.

中文翻译：

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常规无爪图中的权力支配

在本文中，我们首先证明了 $n$ 个顶点上的连通 $4$-regular-free 图的幂支配数至多为 $\frac{n+1}{5}$，并且边界是尖锐的。该陈述部分反驳了 Dorbec 等人提出的猜想。在 SIAM J. Discrete Math., 27:1559-1574, 2013 中。然后我们提出了一种动态规划风格的线性时间算法，用于解决树中的加权幂支配问题。