A Polynomial Kernel for Vertex Deletion to the Scattered Class of Proper Interval Graph and Trees
Ashwin Jacob, Arpit Kumar, Diptapriyo Majumdar

TL;DR
This paper introduces a polynomial kernelization for the (Proper-Interval, Tree)-Vertex Deletion problem, reducing the problem size to a polynomial function of the parameter k, specifically O(k^{33}) vertices.
Contribution
It provides the first nontrivial polynomial kernel for the problem, advancing the understanding of kernelization in hereditary graph class vertex deletion.
Findings
Established a polynomial kernel with O(k^{33}) vertices.
Demonstrated fixed-parameter tractability of the problem.
Contributed to the kernelization theory for hereditary graph class problems.
Abstract
Vertex deletion to hereditary graph class is well-studied in parameterized complexity. Vertex deletion to the scattered graph classes has gained attention in recent years. In this paper, we consider (Proper-Interval, Tree)-Vertex Deletion, the input to which is an undirected graph and an integer . The goal is to pick a set of at most vertices such that is a simple graph and every connected component of is a proper interval graph or a tree. When parameterized by the solution size , (Proper-Interval, Tree)-Vertex Deletion has been proved to be fixed-parameter tractable by Jacob et al. [JCSS-2023, FCT-2021]. In this paper, we consider this problem from the perspective of polynomial kernelization. We provide a first nontrivial polynomial kernel for (Proper-Interval, Tree)-Vertex Deletion, with vertices.
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