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Completeness results for Graph Isomorphism (2002)
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BibTeX
@MISC{Jenner02completenessresults,
author = {Birgit Jenner and Johannes Köbler and Pierre McKenzie and Jacobo Torán},
title = {Completeness results for Graph Isomorphism},
year = {2002}
}
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Abstract
We prove that the graph isomorphism problem restricted to trees and to colored graphs with color multiplicities 2 and 3 is many-one complete for several complexity classes within NC². In particular we show that tree isomorphism, when trees are encoded as strings, is NC¹-hard under AC0-reductions. NC¹-completeness thus follows from Buss's NC¹ upper bound. By contrast, we prove that testing isomorphism of two trees encoded as pointer lists is L-complete. Concerning colored graphs we show that the isomorphism problem for graphs with color multiplicities 2 and 3 is complete for symmetric logarithmic space SL under many-one reductions. This result improves the existing upper bounds for the problem. We also show that the graph automorphism problem for colored graphs with color classes of size 2 is equivalent to deciding whether a graph has more than a single connected component and we prove that for color classes of size 3 the graph automorphism problem is contained in SL.
Keyphrases
graph isomorphism completeness result colored graph color multiplicity graph automorphism problem color class isomorphism problem many-one reduction several complexity class nc upper bound tree isomorphism symmetric logarithmic space sl pointer list upper bound nc completeness graph isomorphism problem
