# 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 ≤ 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)· n^O(1)-time algorithm. We also show that it can be solved in 2^O(f k)· n^O(1) time for the feedback vertex number f when i ≥ 2. In contrast, we find that it is W[1]-hard for the treedepth for any integer i ≥ 1. Finally, we show that Biclique-Free Vertex Deletion has a polynomial kernel for every i ≥ 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.

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