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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/
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Datta, Samir ; Nimbhorkar, Prajakta ; Thierauf, Thomas ; Wagner, Fabian

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},
  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: Graph isomorphism, K_{3,3}-free graphs, K_5-free graphs, log-space
Collection: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science
Issue Date: 2009
Date of publication: 14.12.2009

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