计算H-Free图中的子集顶点覆盖

Nick Brettell, Jelle J. Oostveen, Sukanya Pandey, D. Paulusma, E. J. V. Leeuwen
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引用次数: 0

摘要

我们考虑顶点覆盖的一个自然推广:子集顶点覆盖问题,它是决定对于一个图$G=(V,E)$,一个子集$T \subseteq V$和整数$k$,如果$V$有一个大小不超过$k$的子集$S$,使得$S$包含与$T$的顶点相关的每条边的至少一个端点。如果一个图不包含$H$作为诱导子图,则它是$H$自由的。通过证明亚立方(爪形,菱形)自由平面图和$2$-单极图($2P_3$自由弱弦图的一个子类)上的子集顶点覆盖是np完全的,解决了文献中的两个开放问题。我们的结果首次表明,在P $\neq$ NP条件下,子集顶点覆盖比顶点覆盖更难计算。我们还证明了新的多项式时间结果。我们首先给出了图上$G[T]$ H$自由的二分类。也就是说,我们证明了子集顶点覆盖在图$G$上是多项式时间可解的,其中$G[T]$是$H$自由的,如果$H = sP_1 + tP_2$,否则是np完全的。此外,我们还证明了$(sP_1 + P_2 + P_3)$ free图和有界中宽图的子集顶点覆盖是多项式时间可解的。通过将我们的新结果与已知结果相结合,我们得到了$H$ free图上子集顶点覆盖的部分复杂度分类。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Computing Subset Vertex Covers in H-Free Graphs
We consider a natural generalization of Vertex Cover: the Subset Vertex Cover problem, which is to decide for a graph $G=(V,E)$, a subset $T \subseteq V$ and integer $k$, if $V$ has a subset $S$ of size at most $k$, such that $S$ contains at least one end-vertex of every edge incident to a vertex of $T$. A graph is $H$-free if it does not contain $H$ as an induced subgraph. We solve two open problems from the literature by proving that Subset Vertex Cover is NP-complete on subcubic (claw,diamond)-free planar graphs and on $2$-unipolar graphs, a subclass of $2P_3$-free weakly chordal graphs. Our results show for the first time that Subset Vertex Cover is computationally harder than Vertex Cover (under P $\neq$ NP). We also prove new polynomial time results. We first give a dichotomy on graphs where $G[T]$ is $H$-free. Namely, we show that Subset Vertex Cover is polynomial-time solvable on graphs $G$, for which $G[T]$ is $H$-free, if $H = sP_1 + tP_2$ and NP-complete otherwise. Moreover, we prove that Subset Vertex Cover is polynomial-time solvable for $(sP_1 + P_2 + P_3)$-free graphs and bounded mim-width graphs. By combining our new results with known results we obtain a partial complexity classification for Subset Vertex Cover on $H$-free graphs.
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