# Kernelization for Treewidth-2 Vertex Deletion

The Treewidth-2 Vertex Deletion problem asks whether a set of at most t vertices can be removed from a graph, such that the resulting graph has treewidth at most two. A graph has treewidth at most two if and only if it does not contain a K_4 minor. Hence, this problem corresponds to the NP-hard ℱ-Minor Cover problem with ℱ = {K_4}. For any variant of the ℱ-Minor Cover problem where ℱ contains a planar graph, it is known that a polynomial kernel exists. I.e., a preprocessing routine that in polynomial time outputs an equivalent instance of size t^O(1). However, this proof is non-constructive, meaning that this proof does not yield an explicit bound on the kernel size. The {K_4}-Minor Cover problem is the simplest variant of the ℱ-Minor Cover problem with an unknown kernel size. To develop a constructive kernelization algorithm, we present a new method to decompose graphs into near-protrusions, such that near-protrusions in this new decomposition can be reduced using elementary reduction rules. Our method extends the `approximation and tidying' framework by van Bevern et al. [Algorithmica 2012] to provide guarantees stronger than those provided by both this framework and a regular protrusion decomposition. Furthermore, we provide extensions of the elementary reduction rules used by the {K_4, K_2,3}-Minor Cover kernelization algorithm introduced by Donkers et al. [IPEC 2021]. Using the new decomposition method and reduction rules, we obtain a kernel consisting of O(t^41) vertices, which is the first constructive kernel. This kernel is a step towards more concrete kernelization bounds for the ℱ-Minor Cover problem where ℱ contains a planar graph, and our decomposition provides a potential direction to achieve these new bounds.

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