When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2009.2314
URN: urn:nbn:de:0030-drops-23144
URL: http://drops.dagstuhl.de/opus/volltexte/2009/2314/
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### Graph Isomorphism for K_{3,3}-free and K_5-free graphs is in Log-space

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### Abstract

Graph isomorphism is an important and widely studied computational problem with a yet unsettled complexity. However, the exact complexity is known for isomorphism of various classes of graphs. Recently, \cite{DLNTW09} proved that planar isomorphism is complete for log-space. We extend this result %of \cite{DLNTW09} further to the classes of graphs which exclude \$K_{3,3}\$ or \$K_5\$ as a minor, and give a log-space algorithm. Our algorithm decomposes \$K_{3,3}\$ minor-free graphs into biconnected and those further into triconnected components, which are known to be either planar or \$K_5\$ components \cite{Vaz89}. This gives a triconnected component tree similar to that for planar graphs. An extension of the log-space algorithm of \cite{DLNTW09} can then be used to decide the isomorphism problem. For \$K_5\$ minor-free graphs, we consider \$3\$-connected components. These are either planar or isomorphic to the four-rung mobius ladder on \$8\$ vertices or, with a further decomposition, one obtains planar \$4\$-connected components \cite{Khu88}. We give an algorithm to get a unique decomposition of \$K_5\$ minor-free graphs into bi-, tri- and \$4\$-connected components, and construct trees, accordingly. Since the algorithm of \cite{DLNTW09} does not deal with four-connected component trees, it needs to be modified in a quite non-trivial way.

### BibTeX - Entry

```@InProceedings{datta_et_al:LIPIcs:2009:2314,
author =	{Samir Datta and Prajakta Nimbhorkar and Thomas Thierauf and Fabian Wagner},
title =	{{Graph Isomorphism for K_{3,3}-free and K_5-free graphs is in Log-space}},
booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
pages =	{145--156},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-13-2},
ISSN =	{1868-8969},
year =	{2009},
volume =	{4},
editor =	{Ravi Kannan and K. Narayan Kumar},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},