A Polynomial Kernel for Vertex Deletion to the Scattered Class of Proper Interval Graph and Trees

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 $G = (V, E)$ and an integer $k$. The goal is to pick a set $X \subseteq V(G)$ of at most $k$ vertices such that $G - X$ is a simple graph and every connected component of $G - X$ is a proper interval graph or a tree. When parameterized by the solution size $k$, (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 $O(k^{33})$ vertices.