Abstract
In the ℱMinorFree 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 GS contains no graph from the finite family ℱ as a minor. It is known that whenever ℱ contains at least one planar graph, then ℱMinorFree Deletion admits a polynomial kernel, that is, there is a polynomialtime algorithm that outputs an equivalent instance of size k^{𝒪(1)} [Fomin, Lokshtanov, Misra, Saurabh; FOCS 2012]. However, this result relies on nonconstructive arguments based on wellquasiordering 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 ℱMinorFree Deletion for the family ℱ = {K₄, K_{2,3}}. 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 minorminimal obstruction to having an outerplanar deletion set of size k has 𝒪(k⁴) vertices and edges.
BibTeX  Entry
@InProceedings{donkers_et_al:LIPIcs.IPEC.2021.14,
author = {Donkers, Huib and Jansen, Bart M. P. and W{\l}odarczyk, Micha{\l}},
title = {{Preprocessing for Outerplanar Vertex Deletion: An Elementary Kernel of Quartic Size}},
booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)},
pages = {14:114:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772167},
ISSN = {18688969},
year = {2021},
volume = {214},
editor = {Golovach, Petr A. and Zehavi, Meirav},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/15397},
URN = {urn:nbn:de:0030drops153979},
doi = {10.4230/LIPIcs.IPEC.2021.14},
annote = {Keywords: fixedparameter tractability, kernelization, outerplanar graphs}
}
Keywords: 

fixedparameter tractability, kernelization, outerplanar graphs 
Collection: 

16th International Symposium on Parameterized and Exact Computation (IPEC 2021) 
Issue Date: 

2021 
Date of publication: 

22.11.2021 