Unified almost linear kernels for generalized covering and packing problems on nowhere dense classes

Abstract

Let F be a family of graphs and r≥0 be an integer. For a graph G and an integer k, (r,F)-COVERING asks whether there is a set D⊆V(G) of size at most k such that every induced subgraph of G isomorphic to a graph in F is at distance at most r from D. (r,F)-PACKING asks whether G has k induced subgraphs H1,…,Hk such that each Hi is isomorphic to a graph in F and the distance between distinct V(Hi) and V(Hj) in G is more than r. We show that for every fixed nonempty finite family F of connected graphs and r≥0, (r,F)-COVERING and (r,F)-PACKING 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. (2016) [17] and Eickmeyer et al. (2017) [20] and for DISTANCE-r INDEPENDENT SET by Pilipczuk and Siebertz (2021) [41].