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 CVD of size
O(k161log58k), and asked whether one can design a kernel of size
O(k10). While we do not completely resolve this question, we design a
significantly smaller kernel of size O(k12log10k), inspired by the
O(k2)-size kernel for Feedback Vertex Set. Furthermore, we introduce the
notion of the independence degree of a vertex, which is our main conceptual
contribution