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Graph Isomorphism

1980; McKay

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Encyclopedia of Algorithms

Keywords and Synonyms

Graph matching; Symmetry group      

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.

Formal Description

A 

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Recommended Reading

  1. Babai, L., Luks, E.: Canonical labelling of graphs. In: Proceedings of the 15th Annual ACM Symposium on Theory of Computing, pp. 171–183. ACM, New York (1983)

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  2. Darga, P.T., Liffiton, M.H., Sakallah, K.A., Markov, I.L.: Exploiting Structure in Symmetry Generation for CNF. In: Proceedings of the 41st Design Automation Conference, 2004, pp. 530–534. Source code at http://vlsicad.eecs.umich.edu/BK/SAUCY/

  3. Köbler, J., Schöning, U., Torán, J.: The Graph Isomorphism Problem: its structural complexity. Birkhäuser, Boston (1993)

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  4. McKay, B.D.: Hadamard equivalence via graph isomorphism. Discret. Math. 27, 213–214 (1979)

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  5. McKay, B.D.: Practical graph isomorphism. Congr. Numer. 30, 45–87 (1981)

    MathSciNet  Google Scholar 

  6. McKay, B.D., Meynert, A., Myrvold, W.: Small Latin squares, quasigroups and loops. J. Comb. Des. 15, 98–119 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Miyazaki, T.: The complexity of McKay's canonical labelling algorithm. In: Groups and Computation, II. DIMACS Ser. Discret. Math. Theor. Comput. Sci., vol. 28, pp. 239–256. American Mathematical Society, Providence, RI (1997)

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

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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 ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

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© 2008 Springer-Verlag

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McKay, B. (2008). Graph Isomorphism. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_172

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