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Exploring the Kernelization Borders for Hitting Cycles Agrawal, Akanksha Jain, Pallavi Kanesh, Lawqueen Misra, Pranabendu Saurabh, Saket Parameterized Complexity Kernelization Conflict-free problems Feedback Vertex Set Even Cycle Transversal Odd Cycle Transversal A generalization of classical cycle hitting problems, called conflict version of the problem, is defined as follows. An input is undirected graphs G and H on the same vertex set, and a positive integer k, and the objective is to decide whether there exists a vertex subset X subseteq V(G) such that it intersects all desired "cycles" (all cycles or all odd cycles or all even cycles) and X is an independent set in H. In this paper we study the conflict version of classical Feedback Vertex Set, and Odd Cycle Transversal problems, from the view point of kernelization complexity. In particular, we obtain the following results, when the conflict graph H belongs to the family of d-degenerate graphs. 1) CF-FVS admits a O(k^{O(d)}) kernel. 2) CF-OCT does not admit polynomial kernel (even when H is 1-degenerate), unless NP subseteq coNP/poly. For our kernelization algorithm we exploit ideas developed for designing polynomial kernels for the classical Feedback Vertex Set problem, as well as, devise new reduction rules that exploit degeneracy crucially. Our main conceptual contribution here is the notion of "k-independence preserver". Informally, it is a set of "important" vertices for a given subset X subseteq V(H), that is enough to capture the independent set property in H. We show that for d-degenerate graph independence preserver of size k^{O(d)} exists, and can be used in designing polynomial kernel. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Akanksha Agrawal and Pallavi Jain and Lawqueen Kanesh and Pranabendu Misra and Saket Saurabh 2019 Is Part Of LIPIcs, Volume 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.IPEC.2018.14 urn:nbn:de:0030-drops-102158 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.14 eng https://creativecommons.org/licenses/by/3.0/legalcode