# Introducing lop-kernels: the case of Maximum Minimal Vertex Cover

**Speaker**: Ignasi Sau (LIRMM)

**Abstract:**
In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph G and a positive integer k, and the objective is to decide whether G contains a minimal vertex cover of size at least k. Motivated by the kernelization of MMVC with parameter k, we introduce a simple general framework to obtain lower bounds on the degrees of a certain type of polynomial kernels for vertex optimization problems, which we call "lop-kernels". Informally, this type of kernels is required to preserve large optimal solutions in the reduced instance, and captures the vast majority of existing kernels in the literature. As a consequence of this framework, we show that the trivial quadratic kernel for MMVC is essentially optimal, answering a question of Boria et al. [Discret. Appl. Math. 2015], and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal. On the positive side, given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs, we provide subquadratic kernels on H-free graphs for several graphs H, by making use of the Erdös-Hajnal property. Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, under reasonable complexity assumptions. This indicates that parameters smaller than the solution size are unlikely to yield polynomial kernels for MMVC.

(Joint work with Julio Araújo, Marin Bougeret and Victor A. Campos, available at arXiv/2102.02484).

**Date and time:** 16/09/2021 at 14:30 (Paris time)

**Link to the talk:** https://bbb-temp.grenet.fr/b/lou-9x7-69p