Abstract
In this paper we formulate and study the problem of representing groups on graphs. We show that with respect to polynomial time Turing reducibility, both abelian and solvable group representability are all equivalent to graph isomorphism, even when the group is presented as a permutation group via generators. On the other hand, the representability problem for general groups on trees is equivalent to checking, given a group G and n, whether a nontrivial homomorphism from G to S n exists. There does not seem to be a polynomial time algorithm for this problem, in spite of the fact that tree isomorphism has polynomial time algorithms.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Arvind, V., Kurur, P.P.: Graph Isomorphism is in SPP. In: 43rd Annual Symposium of Foundations of Computer Science, pp. 743–750. IEEE, Los Alamitos (2002)
Babai, L., Luks, E.M.: Canonical labeling of graphs. In: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pp. 171–183 (1983)
Furst, M.L., Hopcroft, J.E., Luks, E.M.: Polynomial-time algorithms for permutation groups. In: IEEE Symposium on Foundations of Computer Science, pp. 36–41 (1980)
Hall Jr., M.: The Theory of Groups, 1st edn. The Macmillan Company, New York (1959)
Joyner, W.D.: Real world applications of representation theory of non-abelian groups, http://www.usna.edu/Users/math/wdj/repn_thry_appl.htm
Köbler, J., Schöning, U., Torán, J.: Graph isomorphism is low for PP. Computational Complexity 2(4), 301–330 (1992)
Mathon, R.: A note on graph isomorphism counting problem. Information Processing Letters 8(3), 131–132 (1979)
Schöning, U.: Graph isomorphism is in the low hierarchy. In: Symposium on Theoretical Aspects of Computer Science, pp. 114–124 (1987)
Sims, C.C.: Computational methods in the study of permutation groups. Computational problems in Abstract Algebra, 169–183 (1970)
Sims, C.C.: Some group theoretic algorithms. Topics in Algebra 697, 108–124 (1978)
Wielandt, H.: Finite Permutation Groups. Academic Press, New York (1964)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dutta, S., Kurur, P.P. (2009). Representing Groups on Graphs. In: Královič, R., Niwiński, D. (eds) Mathematical Foundations of Computer Science 2009. MFCS 2009. Lecture Notes in Computer Science, vol 5734. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03816-7_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-03816-7_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03815-0
Online ISBN: 978-3-642-03816-7
eBook Packages: Computer ScienceComputer Science (R0)