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DOI: 10.4230/LIPIcs.FSTTCS.2009.2314
URN: urn:nbn:de:0030-drops-23144
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2009/2314/
Datta, Samir ;
Nimbhorkar, Prajakta ;
Thierauf, Thomas ;
Wagner, Fabian
Graph Isomorphism for K_{3,3}-free and K_5-free graphs is in Log-space
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},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2009/2314},
URN = {urn:nbn:de:0030-drops-23144},
doi = {10.4230/LIPIcs.FSTTCS.2009.2314},
annote = {Keywords: Graph isomorphism, K_{3,3}-free graphs, K_5-free graphs, log-space}
}
Keywords: |
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Graph isomorphism, K_{3,3}-free graphs, K_5-free graphs, log-space |
Collection: |
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IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science |
Issue Date: |
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2009 |
Date of publication: |
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14.12.2009 |