Unified Almost Linear Kernels for Generalized Covering and Packing Problems on Nowhere Dense Classes

Abstract

Let F be a family of graphs, and let p, r be nonnegative integers. For a graph G and an integer k, the pp, r, Fq-Covering problem asks whether there is a set D Ď V pGq of size at most k such that if the p-th power of G has an induced subgraph isomorphic to a graph in F, then it is at distance at most r from D. The pp, r, Fq-Packing problem asks whether Gp has k induced subgraphs H1, . . ., Hk such that each Hi is isomorphic to a graph in F, and for i, j P t1, . . ., ku, the distance between V pHiq and V pHjq in G is larger than r. We show that for every fixed nonnegative integers p, r and every fixed nonempty finite family F of connected graphs, pp, r, Fq-Covering with p ď 2r ` 1 and pp, r, Fq-Packing with p ď 2tr{2u ` 1 admit almost linear kernels on every nowhere dense class of graphs, parameterized by the solution size k. As corollaries, we prove that Distance-r Vertex Cover, Distance-r Matching, F-Free Vertex Deletion, and Induced-F-Packing for any fixed finite family F of connected graphs admit almost linear kernels on every nowhere dense class of graphs. Our results extend the results for Distance-r Dominating Set by Drange et al. (STACS 2016) and Eickmeyer et al. (ICALP 2017), and for Distance-r Independent Set by Pilipczuk and Siebertz (EJC 2021).