For a non-negative integer $\ell$, a graph $G$ is an $\ell$-leaf power of a tree $T$ if $V(G)$ is equal to the set of leaves of $T$, and distinct vertices $v$ and $w$ of $G$ are adjacent if and only if the distance between $v$ and $w$ in $T$ is at most $\ell$. Given a graph $G$, 3-Leaf Power Deletion asks whether there is a set $S\subseteq V(G)$ of size at most $k$ such that $G\setminus S$ is a $3$-leaf power of some tree $T$. We provide a polynomial kernel for this problem. More specifically, we present a polynomial-time algorithm for an input instance $(G,k)$ to output an equivalent instance $(G',k')$ such that $k'\leq k$ and $G'$ has at most $O(k^{14})$ vertices.

中文翻译：

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$3$-leaf power 删除的多项式核

对于非负整数 $\ell$，如果 $V(G)$ 等于 $T$ 的叶子集合，则图 $G$ 是树 $T$ 的 $\ell$-叶子幂，并且 $G$ 的不同顶点 $v$ 和 $w$ 相邻当且仅当 $T$ 中 $v$ 和 $w$ 之间的距离至多为 $\ell$。给定一个图 $G$，3-Leaf Power Deletion 询问是否存在一个大小至多为 $k$ 的集合 $S\subseteq V(G)$ 使得 $G\setminus S$ 是一个 $3$-leaf Power一些树 $T$。我们为这个问题提供了一个多项式核。更具体地说，我们提出了一个多项式时间算法，用于输入实例 $(G,k)$ 以输出等效实例 $(G',k')$，使得 $k'\leq k$ 和 $G'$ 具有至多 $O(k^{14})$ 个顶点。