The first combinatorial algorithm for the minimum weighted clique cover (MWCC) in a claw‐free perfect graph $G$ due to Hsu and Nemhauser dates back to 1984. It is essentially a “dual” algorithm as it relies on any algorithm for the maximum weighted stable set (MWSS) problem in claw‐free graphs and, taking into account the best‐known complexity for the latter problem, its complexity is $O\left(\mid V\left(G\right){\mid }^5\right)$. More recently, Chudnovsky and Seymour introduced a composition operation, strip composition, to define their structural results for claw‐free graphs; however, this composition operation is general and applies to nonclaw‐free graphs as well. In this paper, we show that an MWCC of a perfect strip‐composed graph, with the basic graphs belonging to a class $\mathcal{G}$, can be found in polynomial time, provided that the MWCC problem can be solved on $\mathcal{G}$ in polynomial time. For the case of claw‐free perfect strip‐composed graphs, the algorithm can be tailored so that it never requires the computation of an MWSS on the strips and can be implemented as to run in $O\left(\mid V\left(G\right){\mid }^3\right)$ time. Finally, building upon several results from the literature, we show how to deal with nonstrip‐composed claw‐free perfect graphs, and therefore compute an MWCC in a general claw‐free perfect graph in $O\left(\mid V\left(G\right){\mid }^3\right)$ time.

中文翻译：

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无爪完美图形上的最小加权派系覆盖

无爪完美图中最小加权派系覆盖（MWCC）的第一个组合算法 $G$ 由于Hsu和Nemhauser的研究可以追溯到1984年。它本质上是“双重”算法，因为它依赖于任何算法来解决无爪图中的最大加权稳定集（MWSS）问题，并考虑到了最著名的复杂性。后一个问题，其复杂性是 $Ø（\mid V（G）{\mid }^5）$。最近，Chudnovsky和Seymour引入了合成操作，带状合成，以定义其无爪图的结构结果。但是，这种合成操作是通用的，也适用于非爪图。在本文中，我们证明了理想的带状图的MWCC，基本图属于一类$\mathcal{G}$可以在多项式时间内找到，前提是可以解决MWCC问题 $\mathcal{G}$在多项式时间内 对于无爪的完美条形图，可以对算法进行定制，以使其永远不需要在条带上计算MWSS，并且可以实现为$Ø（\mid V（G）{\mid }^3）$时间。最后，基于文献的一些结果，我们展示了如何处理非条带组成的无爪完美图，并因此计算了通用无爪完美图中的MWCC。$Ø（\mid V（G）{\mid }^3）$ 时间。