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 subset of 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)) [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 [Thomasse, TALC 2010]. Furthermore, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution.