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

2026-05-04

AI Summary

This study addresses the (Proper-Interval, Tree)-Vertex Deletion problem, which asks whether at most $k$ vertices can be removed from an undirected graph so that each connected component of the remaining graph is either a proper interval graph or a tree. Although this problem is known to be fixed-parameter tractable, no polynomial kernel was previously known. By integrating parameterized algorithmic techniques, graph decomposition methods, and structural characterizations of proper interval graphs and trees, this work devises a suite of efficient reduction rules and constructs, for the first time, a polynomial kernel of size $O(k^{33})$. This result resolves a long-standing barrier in kernelization for this problem and provides a theoretically sound preprocessing guarantee for vertex deletion problems targeting scattered graph classes.

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.

Problem

• Vertex Deletion
• Proper Interval Graph
• Tree
• Polynomial Kernel
• Scattered Graph Class

Innovation

• polynomial kernel
• vertex deletion
• proper interval graph
• tree
• parameterized complexity