In this work, we study the Biclique-Free Vertex Deletion problem: Given a graph $G$ and integers $k$ and $i \le j$, find a set of at most $k$ vertices that intersects every (not necessarily induced) biclique $K_{i, j}$ in $G$. This is a natural generalization of the Bounded-Degree Deletion problem, wherein one asks whether there is a set of at most $k$ vertices whose deletion results in a graph of a given maximum degree $r$. The two problems coincide when $i = 1$ and $j = r + 1$. We show that Biclique-Free Vertex Deletion is fixed-parameter tractable with respect to $k + d$ for the degeneracy $d$ by developing a $2^{O(d k^2)} \cdot n^{O(1)}$-time algorithm. We also show that it can be solved in $2^{O(f k)} \cdot n^{O(1)}$ time for the feedback vertex number $f$ when $i \ge 2$. In contrast, we find that it is W[1]-hard for the treedepth for any integer $i \ge 1$. Finally, we show that Biclique-Free Vertex Deletion has a polynomial kernel for every $i \ge 1$ when parameterized by the feedback edge number. Previously, for this parameter, its fixed-parameter tractability for $i = 1$ was known [Betzler et al., DAM '12] but the existence of polynomial kernel was open.

翻译：在本工作中，我们研究双团自由顶点删除问题：给定图 $G$ 及整数 $k$ 和 $i \le j$，寻找一个最多包含 $k$ 个顶点的集合，使其与 $G$ 中每个（不一定是诱导的）双团 $K_{i, j}$ 相交。这是有界度删除问题的自然推广，后者要求判断是否存在最多 $k$ 个顶点的集合，使得删除这些顶点后得到的图具有给定的最大度 $r$。当 $i = 1$ 且 $j = r + 1$ 时，这两个问题完全等价。我们通过设计 $2^{O(d k^2)} \cdot n^{O(1)}$ 时间算法，证明了双团自由顶点删除问题对于退化度 $d$ 与 $k$ 之和 $k + d$ 是固定参数可解的。我们还证明了当 $i \ge 2$ 时，该问题在反馈顶点数 $f$ 参数下可在 $2^{O(f k)} \cdot n^{O(1)}$ 时间内求解。与之相对，我们发现对于任意整数 $i \ge 1$，该问题在树深参数下是 W[1]-困难的。最后，我们证明了当以反馈边数作为参数时，对于所有 $i \ge 1$，双团自由顶点删除问题存在多项式核。此前针对该参数，仅知 $i = 1$ 时具有固定参数可解性 [Betzler et al., DAM '12]，而多项式核的存在性一直悬而未决。