Interval graphs: canonical representation in logspace
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of 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
A Fourier-theoretic approach for inferring symmetries
Computational Geometry: Theory and Applications
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Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. There is a significant gap between extant lower and upper bounds for planar graphs as well. We bridge the gap for this natural and important special case by presenting an upper bound that matches the known log-space hardness [JKMT03]. In fact, we show the formally stronger result that planar graph canonization is in log-space. This improves the previously known upper bound of AC1 [MR91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to log-space reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in log-space by [DLN08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell’s algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas, including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph. This lemma is crucial for the reduction to work in log-space.