Kawarabayashi, Kenichi ;
Kobayashi, Yusuke
Linear minmax relation between the treewidth of Hminorfree graphs and its largest grid
Abstract
A key theorem in algorithmic graphminor theory is a minmax relation
between the treewidth of a graph and its largest grid minor. This minmax relation is a keystone of the Graph Minor Theory of Robertson
and Seymour, which ultimately proves Wagner's Conjecture about the structure of minorclosed graph properties.
In 2008, Demaine and Hajiaghayi proved a remarkable linear minmax
relation for graphs excluding any fixed minor H: every Hminorfree
graph of treewidth at least c_H r has an r times rgrid minor for some
constant c_H. However, as they pointed out, there is still a major
problem left in this theorem. The problem is that their proof heavily
depends on Graph Minor Theory, most of which lacks explicit bounds and
is believed to have very large bounds. Hence c_H is not explicitly
given in the paper and therefore this result is usually not strong
enough to derive efficient algorithms.
Motivated by this problem, we give another (relatively short and
simple) proof of this result without using big machinery of Graph
Minor Theory. Hence we can give an explicit bound for c_H (an exponential function of a polynomial of H). Furthermore, our result
gives a constant w=2^O(r^2 log r) such that every graph of treewidth
at least w has an r times rgrid minor, which improves the previously
known best bound 2^Theta(r^5)$ given by Robertson, Seymour, and Thomas
in 1994.
BibTeX  Entry
@InProceedings{kawarabayashi_et_al:LIPIcs:2012:3416,
author = {Kenichi Kawarabayashi and Yusuke Kobayashi},
title = {{Linear minmax relation between the treewidth of Hminorfree graphs and its largest grid}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {278289},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897354},
ISSN = {18688969},
year = {2012},
volume = {14},
editor = {Christoph D{\"u}rr and Thomas Wilke},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2012/3416},
URN = {urn:nbn:de:0030drops34165},
doi = {http://dx.doi.org/10.4230/LIPIcs.STACS.2012.278},
annote = {Keywords: grid minor, treewidth, graph minor}
}
Keywords: 

grid minor, treewidth, graph minor 
Seminar: 

29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

Issue date: 

2012 
Date of publication: 

24.02.2012 