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 logspace.
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 logspace algorithm.
Our algorithm decomposes $K_{3,3}$ minorfree 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 logspace algorithm of \cite{DLNTW09}
can then be used to decide the isomorphism problem.
For $K_5$ minorfree graphs, we consider $3$connected components.
These are either planar or isomorphic to the fourrung 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 $K_5$ minorfree graphs into bi, tri and $4$connected components,
and construct trees, accordingly.
Since the algorithm of \cite{DLNTW09} does
not deal with fourconnected component trees, it needs to be modified in a quite nontrivial 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_5free graphs is in Logspace}},
booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
pages = {145156},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897132},
ISSN = {18688969},
year = {2009},
volume = {4},
editor = {Ravi Kannan and K. Narayan Kumar},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2009/2314},
URN = {urn:nbn:de:0030drops23144},
doi = {http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2009.2314},
annote = {Keywords: Graph isomorphism, K_{3,3}free graphs, K_5free graphs, logspace}
}
Keywords: 

Graph isomorphism, K_{3,3}free graphs, K_5free graphs, logspace 
Seminar: 

IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science 
Issue Date: 

2009 
Date of publication: 

14.12.2009 