, Volume 29, Issue 4, pp 1426-1481

Graph isomorphism problem

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Abstract

The article is a creative compilation of certain papers devoted to the graph isomorphism problem, which have appeared in recent years. An approach to the isomorphism problem is proposed in the first chapter, combining, mainly, the works of Babai and Luks. This approach, being to the survey's authors the most promising and fruitful of results, has two characteristic features: the use of information on the special structure of the automorphism group and the profound application of the theory of permutation groups. In particular, proofs are given of the recognizability of the isomorphism of graphs with bounded valences in polynomial time and of all graphs in moderately exponential time. In the second chapter a free exposition is given of the Filotti-Mayer-Miller results on the isomorphism of graphs of bounded genus. New and more complete proofs of the main assertions are presented, as well as an algorithm for the testing of the isomorphism of graphs of genus g in time O(vO(g)), where v is the number of vertices. In the third chapter certain extended means of the construction of algorithms testing an isomorphism are discussed together with probabilistically estimated algorithms and the Las Vegas algorithms. In the fourth chapter the connections of the graph isomorphism problem with other problems are examined.

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 118, pp. 83–158, 1982.