Kernelization dichotomies for hitting minors under structural parameterizations

Abstract

For a finite collection of connected graphs $\mathcal{F}$, the $\mathcal{F}$-MINOR-DELETION problem consists in, given a graph $G$ and an integer $\ell$, deciding whether $G$ contains a vertex set of size at most $\ell$ whose removal results in an $\mathcal{F}$-minor-free graph. We lift the existence of (approximate) polynomial kernels for $\mathcal{F}$-MINOR-DELETION by the solution size to (approximate) polynomial kernels parameterized by the vertex-deletion distance to graphs of bounded elimination distance to $\mathcal{F}$-minor-free graphs. This results in exact polynomial kernels for every family $\mathcal{F}$ that contains a planar graph, and an approximate polynomial kernel for PLANAR VERTEX DELETION. Moreover, combining our result with a previous lower bound, we obtain the following infinite set of dichotomies, assuming $NP⊈coNP/poly$: for any finite set $\mathcal{F}$ of biconnected graphs on at least three vertices containing a planar graph, and any minor-closed class of graphs $\mathcal{C}$, $\mathcal{F}$-MINOR-DELETION admits a polynomial kernel parameterized by the vertex-deletion distance to $\mathcal{C}$ if and only if $\mathcal{C}$ has bounded elimination distance to $\mathcal{F}$-minor-free graphs. For instance, this yields dichotomies for CACTUS VERTEX DELETION, OUTERPLANAR VERTEX DELETION, and TREEWIDTH-$t$ VERTEX DELETION for every integer $t\ge 0$. Prior to our work, such dichotomies were only known for the particular cases of VERTEX COVER and FEEDBACK VERTEX SET. We also provide lower bounds on the size of the kernels.

76 pages, 20 figures. An extended abstract presenting the kernelization results of this paper appeared in the Proceedings of STACS 2026, volume 364 of LIPIcs, pages 17:1--17:19. This extended abstract did not contain the lower bounds presented in Section 7 and Appendix B of this version, nor the analysis of the size of the kernels presented in Appendix A