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

Let ℱ be a family of graphs, and let p,r be nonnegative integers. The (p,r,ℱ)-Covering problem asks whether for a graph G and an integer k, there exists a set D of at most k vertices in G such that G^p∖ N_G^r[D] has no induced subgraph isomorphic to a graph in ℱ, where G^p is the p-th power of G. The (p,r,ℱ)-Packing problem asks whether for a graph G and an integer k, G^p has k induced subgraphs H_1,…,H_k such that each H_i is isomorphic to a graph in ℱ, and for distinct i,j∈{1, …, k}, the distance between V(H_i) and V(H_j) 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, the (p,r,ℱ)-Covering problem with p≤2r+1 and the (p,r,ℱ)-Packing problem with p≤2⌊ r/2⌋+1 admit almost linear kernels on every nowhere dense class of graphs, and admit linear kernels on every class of graphs with bounded expansion, parameterized by the solution size k. We obtain the same kernels for their annotated variants. 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 and linear kernels on every class of graphs with bounded expansion. Our results extend the results for Distance-r Dominating Set by Drange et al. (STACS 2016) and Eickmeyer et al. (ICALP 2017), and the result for Distance-r Independent Set by Pilipczuk and Siebertz (EJC 2021).

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