In this work we study a classic generalization of the ubiquitous Vertex Cover (VC) problem, called the Component Order Connectivity (COC) problem. In COC, given an undirected graph G, integers d ≥ 1 and k, the goal is to determine if there is a set of at most k vertices whose deletion results in a graph where each connected component has at most d vertices. When d = 1, this is exactly VC. This work is inspired by polynomial kernelization results with respect to structural parameters for VC. On one hand, Jansen & Bodlaender [TOCS 2013] show that VC admits a polynomial kernel when the parameter is the distance to treewidth-1 graphs, on the other hand Cygan, Lokshtanov, Pilipczuk, Pilipczuk & Saurabh [TOCS 2014] showed that VC does not admit a polynomial kernel when the parameter is distance to treewidth-2 graphs. Greilhuber & Sharma [IPEC 2024] showed that, for any d ≥ 2, d-COC cannot admit a polynomial kernel when the parameter is distance to a forest of pathwidth 2. Here, d-COC is the variant of COC where d is a fixed constant rather than part of the input. We complement this result and show that, analogously to the VC setting, where distance to treewidth-1 graphs versus distance to treewidth-2 graphs is the dividing line between structural parameterizations that admit and respectively do not admit polynomial kernelization, for COC this dividing line lies between distance to pathwidth-1 graphs and distance to pathwidth-2 graphs. The main technical result of this work is that COC admits a polynomial kernel parameterized by distance to pathwidth-1 graphs plus d. The problem d-COC can also be expressed as an ℱ-MinorDeletion problem for an appropriate graph family ℱ. One of the central questions around ℱ-MinorDeletion is for which families ℱ and minor-closed graph classes ???? the problem admits a polynomial kernel when parameterized by the distance to ????. For some families ℱ complete dichotomies answering this question are known [Bougeret et al., SIDMA 2022][Bougeret et al., STACS 2026][Bougeret et al., arXiv 2026]. But, these results do not capture the 2-COC problem. We show that, when d ≥ 2, the line of tractability for polynomial kernelization of d-COC parameterized by the distance to ???? is different from the tractability line of the ℱ-MinorDeletion problems for which the currently known dichotomies apply. Thus, with our result, d-COC serves as an outlier in the class of ℱ-MinorDeletion problems when it comes to understanding the dichotomies for polynomial kernelization when parameterizing by the distance to some minor-closed graph class.
MFCS International Symposium on Mathematical Foundations of Computer Science (MFCS)
2026-08-21
2026-08-29