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：本文研究双团自由顶点删除问题：给定图$G$和整数$k$及$i \le j$，求最多$k$个顶点的集合，使其与$G$中每个（未必是诱导的）双团$K_{i, j}$相交。这是有界度删除问题的自然推广，后者询问是否存在最多$k$个顶点的集合，删除后得到的图具有给定最大度$r$。当$i=1$且$j=r+1$时，两问题重合。我们通过开发$2^{O(d k^2)} \cdot n^{O(1)}$时间算法，证明双团自由顶点删除问题关于退化度$d$的参数$k+d$是固定参数可解的。当$i \ge 2$时，我们还证明该问题可在$2^{O(f k)} \cdot n^{O(1)}$时间内关于反馈顶点数$f$求解。与之相反，我们发现对于任意整数$i \ge 1$，该问题关于树深是W[1]-难的。最后，我们证明当以反馈边数为参数时，双团自由顶点删除问题对每个$i \ge 1$均具有多项式核。此前，对于该参数，已知$i=1$时的固定参数可解性[Betzler等人, DAM '12]，但多项式核的存在性尚未解决。