In graph theory, a branch of mathematics, graph canonization is the problem of finding a canonical form of a given graph G. A canonical form is a labeled graph Canon(G) that is isomorphic to G, such that every graph that is isomorphic to G has the same canonical form as G. Thus, from a solution to the graph canonization problem, one could also solve the problem of graph isomorphism: to test whether two graphs G and H are isomorphic, compute their canonical forms Canon(G) and Canon(H), and test whether these two canonical forms are identical.
The canonical form of a graph is an example of a complete graph invariant: every two isomorphic graphs have the same canonical form, and every two non-isomorphic graphs have different canonical forms.[1][2] Conversely, every complete invariant of graphs may be used to construct a canonical form.[3] The vertex set of an n-vertex graph may be identified with the integers from 1 to n, and using such an identification a canonical form of a graph may also be described as a permutation of its vertices. Canonical forms of a graph are also called canonical labelings,[4] and graph canonization is also sometimes known as graph canonicalization.
^Arvind, Vikraman; Das, Bireswar; Köbler, Johannes (2008), "A logspace algorithm for partial 2-tree canonization", Computer Science – Theory and Applications: Third International Computer Science Symposium in Russia, CSR 2008 Moscow, Russia, June 7-12, 2008, Proceedings, Lecture Notes in Comput. Sci., vol. 5010, Springer, Berlin, pp. 40–51, doi:10.1007/978-3-540-79709-8_8, MR 2475148.
^Arvind, V.; Das, Bireswar; Köbler, Johannes (2007), "The space complexity of k-tree isomorphism", Algorithms and Computation: 18th International Symposium, ISAAC 2007, Sendai, Japan, December 17-19, 2007, Proceedings, Lecture Notes in Comput. Sci., vol. 4835, Springer, Berlin, pp. 822–833, doi:10.1007/978-3-540-77120-3_71, MR 2472661.
^Gurevich, Yuri (1997), "From invariants to canonization" (PDF), Bulletin of the European Association for Theoretical Computer Science (63): 115–119, MR 1621595.
^Babai, László; Luks, Eugene (1983), "Canonical labeling of graphs", Proc. 15th ACM Symposium on Theory of Computing, pp. 171–183, doi:10.1145/800061.808746.
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