Let $H$ be a fixed graph. Given a graph $G$ and an integer $k$, the $H$-free edge modification problem asks whether it is possible to modify at most $k$ edges in $G$ to make it $H$-free. Sandeep and Sivadasan (IPEC 2015) asks whether the paw-free completion problem and the paw-free edge deletion problem admit polynomial kernels. We answer both questions affirmatively by presenting, respectively, $O(k)$-vertex and $O(k^4)$-vertex kernels for them. This is part of an ongoing program that aims at understanding compressibility of $H$-free edge modification problems.

中文翻译：

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用于无爪边缘修改问题的多项式核

令 $H$ 为固定图。给定一个图 $G$ 和一个整数 $k$，无 $H$ 边修改问题询问是否可以修改 $G$ 中至多 $k$ 边以使其无 $H$。Sandeep 和 Sivadasan (IPEC 2015) 询问无爪补全问题和无爪边删除问题是否承认多项式核。我们通过分别为它们呈现 $O(k)$-vertex 和 $O(k^4)$-vertex kernels 来肯定地回答这两个问题。这是一项正在进行的计划的一部分，该计划旨在了解无 $H$ 边缘修改问题的可压缩性。