Canonical labeling of graphs

We announce an algebraic approach to the problem of assigning <italic>canonical forms</italic> to graphs. We compute canonical forms and the associated canonical labelings (or renumberings) in polynomial time for graphs of bounded valence, in moderately exponential, exp(n<supscrpt>½ + &ogr;(1)</supscrpt>),time for general graphs, in subexponential, n<supscrpt>log n</supscrpt>, time for tournaments and for 2-(&ngr;,&kgr;,λ) block designs with &kgr;,λ bounded and n<supscrpt>log log n</supscrpt> time for λ-planes (symmetric designs) with λ bounded. We prove some related problems NP-hard and indicate some open problems.

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