Feedback vertex set inspired kernel for chordal vertex deletion

Given a graph G and a parameter k, the Chordal Vertex Deletion (CVD) problem asks whether there exists a subset U ⊆ 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 1 CVD of size O(k ¹⁶¹ log ⁵⁸ k) and asked whether one can design a kernel of size O(k ¹⁰ ) [Jansen an Pilipczuk, SODA 2017]. 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 [Thomassé, TALG 2010]. Furthermore, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution.