Structural Parameterizations of the Biclique-Free Vertex Deletion Problem

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., 2012) but the existence of polynomial kernel was open.