Does co-NP have short interactive proofs?
Information Processing Letters
Graph isomorphism is in the low hierarchy
Journal of Computer and System Sciences
Isomorphism testing of k-trees is in NC, for fixed k
Information Processing Letters
Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees
Journal of Algorithms
Parallel algorithms for shared-memory machines
Handbook of theoretical computer science (vol. A)
Parallel tree contraction part 2: further applications
SIAM Journal on Computing
A logspace algorithm for tree canonization (extended abstract)
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Proceedings of the 30th IEEE symposium on Foundations of computer science
Automorphism groups, isomorphism, reconstruction
Handbook of combinatorics (vol. 2)
On a New High Dimensional Weisfeiler-Lehman Algorithm
Journal of Algebraic Combinatorics: An International Journal
Isomorphism testing for embeddable graphs through definability
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Definability and Descriptive Complexity on Databases of Bounded Tree-Width
ICDT '99 Proceedings of the 7th International Conference on Database Theory
Moderately Exponential Bound for Graph Isomorphism
FCT '81 Proceedings of the 1981 International FCT-Conference on Fundamentals of Computation Theory
Fixed-Point Logics on Planar Graphs
LICS '98 Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science
Isomorphism of graphs with bounded eigenvalue multiplicity
STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
Linear time algorithm for isomorphism of planar graphs (Preliminary Report)
STOC '74 Proceedings of the sixth annual ACM symposium on Theory of computing
Isomorphism testing for graphs of bounded genus
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Completeness results for graph isomorphism
Journal of Computer and System Sciences
On the Hardness of Graph Isomorphism
SIAM Journal on Computing
How complex are random graphs in first order logic?
Random Structures & Algorithms - Proceedings of the Eleventh International Conference "Random Structures and Algorithms," August 9—13, 2003, Poznan, Poland
The first order definability of graphs with separators via the Ehrenfeucht game
Theoretical Computer Science - Game theory meets theoretical computer science
The first order definability of graphs: upper bounds for quantifier depth
Discrete Applied Mathematics
The Strange Logic of Random Graphs
The Strange Logic of Random Graphs
Graph and map isomorphism and all polyhedral embeddings in linear time
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Planar graphs: logical complexity and parallel isomorphism tests
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
The space complexity of k-tree isomorphism
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
A logspace algorithm for partial 2-tree canonization
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
From invariants to canonization in parallel
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
Fixed-point definability and polynomial time on chordal graphs and line graphs
Fields of logic and computation
Graphs of bounded treewidth can be canonized in AC1
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
Interval Graphs: Canonical Representations in Logspace
SIAM Journal on Computing
The isomorphism problem for k-trees is complete for logspace
Information and Computation
Restricted space algorithms for isomorphism on bounded treewidth graphs
Information and Computation
Fixed-point definability and polynomial time on graphs with excluded minors
Journal of the ACM (JACM)
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Our starting point is the observation that if graphs in a class C have low descriptive complexity, then the isomorphism problem for C is solvable by a fast parallel algorithm. More precisely, we prove that if every graph in C is definable in a finite-variable first order logic with counting quantifiers within logarithmic quantifier depth, then Graph Isomorphism for C is in ${\rm TC^1}\subseteq\mbox{\rm NC\ensuremath{^{2}}}$. If no counting quantifiers are needed, then Graph Isomorphism for C is even in AC1. The definability conditions can be checked by designing a winning strategy for suitable Ehrenfeucht-Fraïssé games with a logarithmic number of rounds. The parallel isomorphism algorithm this approach yields is a simple combinatorial algorithm known as the Weisfeiler-Lehman (WL) algorithm. Using this approach, we prove that isomorphism of graphs of bounded treewidth is testable in TC1, answering an open question from [9]. Furthermore, we obtain an AC1 algorithm for testing isomorphism of rotation systems (combinatorial specifications of graph embeddings). The AC1 upper bound was known before, but the fact that this bound can be achieved by the simple WL algorithm is new. Combined with other known results, it also yields a new AC1 isomorphism algorithm for planar graphs.