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 CVD of size O(k log k), and asked whether one can design a kernel of size O(k). While we do not completely resolve this question, we design a significantly smaller kernel of size O(k log k), inspired by the O(k)-size kernel for Feedback Vertex Set. Furthermore, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution. ∗A preliminary version of this paper appeared in the proceedings of 28th ACM-SIAM Symposium on Discrete Algorithms (SODA 2017). The research leading to these results received funding from the European Research Council under the European Unions Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 306992. †University of Bergen, Bergen, Norway. [email protected]. ‡University of Bergen, Bergen, Norway. [email protected]. §The Institute of Mathematical Sciences, HBNI, Chennai, India. [email protected]. ¶University of Bergen, Bergen, Norway. The Institute of Mathematical Sciences, HBNI, Chennai, India. [email protected]. ‖University of Bergen, Bergen, Norway. [email protected]. ar X iv :1 70 7. 04 91 7v 1 [ cs .D S] 1 6 Ju l 2 01 7