Preprocessing for outerplanar vertex deletion: An elementary kernel of quartic size

Huib Donkers, Bart M.P. Jansen, Michał Włodarczyk

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

In the F-Minor-Free Deletion problem one is given an undirected graph G, an integer k, and the task is to determine whether there exists a vertex set S of size at most k, so that G − S contains no graph from the finite family F as a minor. It is known that whenever F contains at least one planar graph, then F-Minor-Free Deletion admits a polynomial kernel, that is, there is a polynomial-time algorithm that outputs an equivalent instance of size kO(1) [Fomin, Lokshtanov, Misra, Saurabh; FOCS 2012]. However, this result relies on non-constructive arguments based on well-quasi-ordering and does not provide a concrete bound on the kernel size. We study the Outerplanar Deletion problem, in which we want to remove at most k vertices from a graph to make it outerplanar. This is a special case of F-Minor-Free Deletion for the family F = {K4, K2,3}. The class of outerplanar graphs is arguably the simplest class of graphs for which no explicit kernelization size bounds are known. By exploiting the combinatorial properties of outerplanar graphs we present elementary reduction rules decreasing the size of a graph. This yields a constructive kernel with O(k4) vertices and edges. As a corollary, we derive that any minor-minimal obstruction to having an outerplanar deletion set of size k has O(k4) vertices and edges.

Original languageEnglish
Title of host publication16th International Symposium on Parameterized and Exact Computation, IPEC 2021
EditorsPetr A. Golovach, Meirav Zehavi
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772167
DOIs
StatePublished - 1 Nov 2021
Externally publishedYes
Event16th International Symposium on Parameterized and Exact Computation, IPEC 2021 - Virtual, Lisbon, Portugal
Duration: 8 Sep 202110 Sep 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume214
ISSN (Print)1868-8969

Conference

Conference16th International Symposium on Parameterized and Exact Computation, IPEC 2021
Country/TerritoryPortugal
CityVirtual, Lisbon
Period8/09/2110/09/21

Keywords

  • Fixed-parameter tractability
  • Kernelization
  • Outerplanar graphs

ASJC Scopus subject areas

  • Software

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