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Graph Isomorphism

  • Reference work entry
  • First Online:
Encyclopedia of Algorithms

  • 456 Accesses

Years and Authors of Summarized Original Work

  • 1980; McKay

Problem Definition

The problem of determining isomorphism of two combinatorial structures is a ubiquitous one, with applications in many areas. The paradigm case of concern in this chapter is isomorphism of two graphs. In this case, an isomorphism consists of a bijection between the vertex sets of the graphs which induces a bijection between the edge sets of the graphs. One can also take the second graph to be a copy of the first, so that isomorphisms map a graph onto themselves. Such isomorphisms are called automorphisms or, less formally, symmetries. The set of all automorphisms forms a group under function composition called the automorphism group. Computing the automorphism group is a problem rather similar to that of determining isomorphisms.

Graph isomorphism is closely related to many other types of isomorphism of combinatorial structures. In the section entitled “Applications”, several examples are given.

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Recommended Reading

  1. Babai L, Luks E (1983) Canonical labelling of graphs. In: Proceedings of the 15th annual ACM symposium on theory of computing. ACM, New York, pp 171–183

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  3. Köbler J, Schöning U, Torán J (1993) The graph isomorphism problem: its structural complexity. Birkhäuser, Boston

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  8. Toran J (2004) On the hardness of graph isomorphism. SIAM J Comput 33:1093–1108

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Author information

Authors and Affiliations

  1. Department of Computer Science, Australian National University, Canberra, ACT, Australia

    Brendan D. McKay

Authors
  1. Brendan D. McKay
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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McKay, B.D. (2016). Graph Isomorphism. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_172

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