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26
On the Hardness of Graph Isomorphism
 SIAM J. COMPUT
"... We show that the graph isomorphism problem is hard under DLOGTIME uniform AC0 manyone reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class ModkL and for the class DET of problems NC¹ reducible to the determinant. These are the stro ..."
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Cited by 36 (1 self)
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We show that the graph isomorphism problem is hard under DLOGTIME uniform AC0 manyone reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class ModkL and for the class DET of problems NC¹ reducible to the determinant. These are the strongest known hardness results for the graph isomorphism problem and imply a randomized logarithmic space reduction from the perfect matching problem to graph isomorphism. We also investigate hardness results for the graph automorphism problem.
THE ISOMORPHISM PROBLEM FOR PLANAR 3CONNECTED GRAPHS IS IN UNAMBIGUOUS LOGSPACE
, 2008
"... The isomorphism problem for planar graphs is known to be efficiently solvable. For planar 3connected graphs, the isomorphism problem can be solved by efficient parallel algorithms, it is in the class AC¹. In this paper we improve the upper bound for planar 3connected graphs to unambiguous logspace ..."
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Cited by 12 (5 self)
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The isomorphism problem for planar graphs is known to be efficiently solvable. For planar 3connected graphs, the isomorphism problem can be solved by efficient parallel algorithms, it is in the class AC¹. In this paper we improve the upper bound for planar 3connected graphs to unambiguous logspace, in fact to UL ∩ coUL. As a consequence of our method we get that the isomorphism problem for oriented graphs is in NL. We also show that the problems are hard for L.
Planar graph isomorphism is in logspace
 In IEEE Conference on Computational Complexity
, 2009
"... Abstract. We show that the isomorphism of 3connected planar graphs can be decided in deterministic logspace. This improves the previously known bound UL ∩ coUL of [13]. 1 ..."
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Cited by 9 (2 self)
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Abstract. We show that the isomorphism of 3connected planar graphs can be decided in deterministic logspace. This improves the previously known bound UL ∩ coUL of [13]. 1
Algorithm and Experiments in Testing Planar Graphs for Isomorphism
, 2004
"... We give an algorithm for isomorphism testing of planar graphs suitable for practical implementation. The algorithm is based on the decomposition of a graph into biconnected components and further into SPQRtrees. We provide a proof of the algorithm’s correctness and a complexity analysis. We determi ..."
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Cited by 8 (0 self)
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We give an algorithm for isomorphism testing of planar graphs suitable for practical implementation. The algorithm is based on the decomposition of a graph into biconnected components and further into SPQRtrees. We provide a proof of the algorithm’s correctness and a complexity analysis. We determine the conditions in which the implemented algorithm outperforms other graph matchers, which do not impose topological restrictions on graphs. We report experiments with our planar graph matcher tested against McKay’s, Ullmann’s, and SUBDUE’s (a graphbased data mining system) graph matchers.
A Logspace Algorithm for Partial 2Tree canonization
, 2008
"... We show that partial 2tree canonization, and hence isomorphism testing for partial 2trees, is in deterministic logspace. Our algorithm involves two steps: (a) We exploit the “tree of cycles ” property of biconnected partial 2trees to canonize them in logspace. (b) We analyze Lindell’s tree cano ..."
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Cited by 8 (1 self)
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We show that partial 2tree canonization, and hence isomorphism testing for partial 2trees, is in deterministic logspace. Our algorithm involves two steps: (a) We exploit the “tree of cycles ” property of biconnected partial 2trees to canonize them in logspace. (b) We analyze Lindell’s tree canonization algorithm and show that canonizing general partial 2trees is also in logspace, using the algorithm to canonize biconnected partial 2trees.
Bounded Color Multiplicity Graph Isomorphism is in the #L Hierarchy
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 121 (2004)
, 2004
"... In this paper we study the complexity of Bounded Color Multiplicity Graph Isomorphism BCGIb: the input is a pair of vertexcolored graphs such that the number of vertices of a given color in an input graph is bounded by b. We show that BCGIb is in the #L hierarchy (more precisely, the ModkL hierarch ..."
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Cited by 5 (2 self)
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In this paper we study the complexity of Bounded Color Multiplicity Graph Isomorphism BCGIb: the input is a pair of vertexcolored graphs such that the number of vertices of a given color in an input graph is bounded by b. We show that BCGIb is in the #L hierarchy (more precisely, the ModkL hierarchy for some constant k depending on b). Combined with the fact that Bounded Color Multiplicity Graph Isomorphism is logspace manyone hard for every set in the ModkL hierarchy for any constant k, we get a tight classification of the problem using logspacebounded counting classes.
Colored Hypergraph Isomorphism is Fixed Paramter Tractable
 Electronic Colloquium on Computation Complexity
, 2009
"... We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism which has running time 2 O(b) N O(1) , where the parameter b is the maximum size of the color classes of the given hypergraphs and N is the input size. We also describe fpt algorithms for certain permutation g ..."
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Cited by 5 (1 self)
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We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism which has running time 2 O(b) N O(1) , where the parameter b is the maximum size of the color classes of the given hypergraphs and N is the input size. We also describe fpt algorithms for certain permutation group problems that are used as subroutines in our algorithm. Fixed parameter tractability, fpt algorithms, graph isomorphism, com
Restricted space algorithms for isomorphism on bounded treewidth graphs
 IN STACS
, 2010
"... The Graph Isomorphism problem restricted to graphs of bounded treewidth or bounded tree distance width are known to be solvable in polynomial time [2],[19]. We give restricted space algorithms for these problems proving the following results: • Isomorphism for bounded tree distance width graphs is i ..."
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Cited by 4 (0 self)
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The Graph Isomorphism problem restricted to graphs of bounded treewidth or bounded tree distance width are known to be solvable in polynomial time [2],[19]. We give restricted space algorithms for these problems proving the following results: • Isomorphism for bounded tree distance width graphs is in L and thus complete for the class. We also show that for this kind of graphs a canon can be computed within logspace. • For bounded treewidth graphs, when both input graphs are given together with a tree decomposition, the problem of whether there is an isomorphism which respects the decompositions (i.e. considering only isomorphisms mapping bags in one decomposition blockwise onto bags in the other decomposition) is in L. • For bounded treewidth graphs, when one of the input graphs is given with a tree decomposition the isomorphism problem is in LogCFL. • As a corollary the isomorphism problem for bounded treewidth graphs is in LogCFL. This improves the known TC¹ upper bound for the problem given by Grohe and Verbitsky [8].
On graph isomorphism for restricted graph classes
 In
, 2006
"... Abstract. Graph isomorphism (GI) is one of the few remaining problems in NP whose complexity status couldn’t be solved by classifying it as being either NPcomplete or solvable in P. Nevertheless, efficient (polynomialtime or even NC) algorithms for restricted versions of GI have been found over th ..."
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Abstract. Graph isomorphism (GI) is one of the few remaining problems in NP whose complexity status couldn’t be solved by classifying it as being either NPcomplete or solvable in P. Nevertheless, efficient (polynomialtime or even NC) algorithms for restricted versions of GI have been found over the last four decades. Depending on the graph class, the design and analysis of algorithms for GI use tools from various fields, such as combinatorics, algebra and logic. In this paper, we collect several complexity results on graph isomorphism testing and related algorithmic problems for restricted graph classes from the literature. Further, we provide some new complexity bounds (as well as easier proofs of some known results) and highlight some open questions. 1