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Uniform Polynomial Kernel for Deletion to K_{2,p} Minor-Free Graphs

Authors William Lochet , Roohani Sharma

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LIPIcs.ISAAC.2024.46.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Uniform polynomial kernel
  • ℱ-minor-free deletion
  • complete bipartite minor-free graphs
  • K_{2,p}
  • protrusions

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Abstract

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.

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William Lochet and Roohani Sharma. Uniform Polynomial Kernel for Deletion to K_{2,p} Minor-Free Graphs. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.46

Author Details

William Lochet
  • LIRMM, Université de Montpellier, CNRS, Montpellier, France
Roohani Sharma
  • Department of Informatics, University of Bergen, Norway

References

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