10.4230/LIPICS.FSTTCS.2009.2314
Datta, Samir
Samir
Datta
Nimbhorkar, Prajakta
Prajakta
Nimbhorkar
Thierauf, Thomas
Thomas
Thierauf
Wagner, Fabian
Fabian
Wagner
Graph Isomorphism for K_{3,3}-free and K_5-free graphs is in Log-space
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
2009
Article
Graph isomorphism
K_{3
3}-free graphs
K_5-free graphs
log-space
Kannan, Ravi
Ravi
Kannan
Kumar, K. Narayan
K. Narayan
Kumar
2009
2009-12-14
2009-12-14
2009-12-14
en
urn:nbn:de:0030-drops-23144
10.4230/LIPIcs.FSTTCS.2009
978-3-939897-13-2
1868-8969
10.4230/LIPIcs.FSTTCS.2009
LIPIcs, Volume 4, FSTTCS 2009
IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science
2013
4
13
145
156
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Kannan, Ravi
Ravi
Kannan
Kumar, K. Narayan
K. Narayan
Kumar
1868-8969
Leibniz International Proceedings in Informatics (LIPIcs)
2009
4
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
12 pages
123830 bytes
application/pdf
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license
info:eu-repo/semantics/openAccess
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.
LIPIcs, Vol. 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, pages 145-156