无爪编辑的多项式核

E. Eiben, W. Lochet, Saket Saurabh
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引用次数: 9

摘要

对于一个固定的图$H$, $H$自由编辑问题是问我们是否可以通过添加或删除最多$k$条边来修改给定的图$G$,使得结果图不包含$H$作为诱导子图。已知该问题对于所有至少有$3$顶点的固定$H$是np完全的,并且它允许使用$2^{O(k)}n^{O(1)}$算法。Cai和Cai证明了$H$自由编辑问题不承认多项式核,只要$H$或它的补是一个至少有$4$条边的路径或循环,或者是一个至少有$1$条边缺失的$3$连通图。他们的结果表明,如果$H$不是独立集或团,那么$H$-自由编辑只允许对少数小图$H$使用多项式核,除非$\textsf{coNP} \in \textsf{NP/poly}$。因此,解决小图$H$自由编辑的$H$核化问题对于获得该问题的完全二分类起着至关重要的作用。在本文中,我们正面地回答了最后两个未解析图$H$在$4$顶点上的可压缩性问题。也就是说,我们给出了$O(k^{6})$顶点的无爪编辑的第一个多项式核。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A Polynomial Kernel for Paw-Free Editing
For a fixed graph $H$, the $H$-free-editing problem asks whether we can modify a given graph $G$ by adding or deleting at most $k$ edges such that the resulting graph does not contain $H$ as an induced subgraph. The problem is known to be NP-complete for all fixed $H$ with at least $3$ vertices and it admits a $2^{O(k)}n^{O(1)}$ algorithm. Cai and Cai showed that the $H$-free-editing problem does not admit a polynomial kernel whenever $H$ or its complement is a path or a cycle with at least $4$ edges or a $3$-connected graph with at least $1$ edge missing. Their results suggest that if $H$ is not independent set or a clique, then $H$-free-editing admits polynomial kernels only for few small graphs $H$, unless $\textsf{coNP} \in \textsf{NP/poly}$. Therefore, resolving the kernelization of $H$-free-editing for small graphs $H$ plays a crucial role in obtaining a complete dichotomy for this problem. In this paper, we positively answer the question of compressibility for one of the last two unresolved graphs $H$ on $4$ vertices. Namely, we give the first polynomial kernel for paw-free editing with $O(k^{6})$vertices.
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