Feedback Vertex Set Inspired Kernel for Chordal Vertex Deletion

TL;DR

This paper presents a new polynomial kernel for the Chordal Vertex Deletion problem, inspired by Feedback Vertex Set, with a size of O(k^{12} log^{10}k), and introduces the concept of vertex independence degree.

Contribution

It introduces the independence degree of a vertex and designs a smaller kernel for CVD, advancing the understanding of kernelization in parameterized complexity.

Findings

Developed a kernel of size O(k^{12} log^{10}k) for CVD.

Introduced the concept of vertex independence degree.

Connected kernelization techniques to Feedback Vertex Set.

Abstract

Given a graph $G$ and a parameter $k$, the Chordal Vertex Deletion (CVD) problem asks whether there exists a subset $U\subseteq V(G)$ of size at most $k$ that hits all induced cycles of size at least 4. The existence of a polynomial kernel for CVD was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomial kernel for CVD of size $O(k^{161}\log^{58}k)$, and asked whether one can design a kernel of size $O(k^{10})$. While we do not completely resolve this question, we design a significantly smaller kernel of size $O(k^{12}\log^{10}k)$, inspired by the $O(k^2)$-size kernel for Feedback Vertex Set. Furthermore, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution.