2026-08-07T13:42:14Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:22173 2024-12-04T06:53:44Z ddc:004 open_access

Uniform Polynomial Kernel for Deletion to K_{2,p} Minor-Free Graphs Lochet, William Sharma, Roohani Uniform polynomial kernel ℱ-minor-free deletion complete bipartite minor-free graphs K_{2,p} protrusions In the F-Deletion problem, where F is a fixed finite family of graphs, the input is a graph G and an integer k, and the goal is to determine if there exists a set of at most k vertices whose deletion results in a graph that does not contain any graph of F as a minor. The F-Deletion problem encapsulates a large class of natural and interesting graph problems like Vertex Cover, Feedback Vertex Set, Treewidth-η Deletion, Treedepth-η Deletion, Pathwidth-η Deletion, Outerplanar Deletion, Vertex Planarization and many more. We study the F-Deletion problem from the kernelization perspective. In a seminal work, Fomin et al. [FOCS 2012] gave a polynomial kernel for this problem when the family F contains at least one planar graph. The asymptotic growth of the size of the kernel is not uniform with respect to the family F: that is, the size of the kernel is k^{f(F)}, for some function f that depends only on F. Later Giannopoulou et al. [TALG 2017] showed that the non-uniformity in the kernel size bound is unavoidable as Treewidth-η Deletion cannot admit a kernel of size 𝒪(k^{(η+1)/2 - ε}), for any ε > 0, unless NP ⊆ coNP/poly. On the other hand it was also shown that Treedepth-η Deletion admits a uniform kernel of size f(F) ⋅ k⁶ depicting that there are subclasses of F where the asymptotic kernel sizes do not grow as a function of the family F. This work led to the question of determining classes of F where the problem admits uniform polynomial kernels. In this paper, we show that if all the graphs in F are connected and ℱ contains K_{2,p} (a bipartite graph with 2 vertices on one side and p vertices on the other), then the problem admits a uniform kernel of size f(F) ⋅ k^10. The graph K_{2,p} is one natural extension of the graph θ_p, where θ_p is a graph on two vertices and p parallel edges. The case when F contains θ_p has been studied earlier and serves as (the only) other example where the problem admits a uniform polynomial kernel. Schloss Dagstuhl – Leibniz-Zentrum für Informatik William Lochet and Roohani Sharma 2024 Is Part Of LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ISAAC.2024.46 urn:nbn:de:0030-drops-221731 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.46 eng https://creativecommons.org/licenses/by/4.0/legalcode