Abstract
To locally complement a simple graphF at one of its verticesv is to replace the subgraph induced byF onn(v)={w:vw is an edge ofF} by the complementary subgraph. Graphs related by a sequence of local complementations are said to be locally equivalent. We associate a system of equations with unknowns inGF(2) to any pair of graphs {F, F′}, so thatF is locally equivalent toF′ if and only if the system has a solution. The equations are either linear and homogenous or bilinear, and we find a solution, if any, in polynomial time.
Similar content being viewed by others
References
L. Ally: personal communication, May 1988.
A. Bouchet: Isotropic systems,European J. of Combinatorics,8 (1987) 231–244.
A. Bouchet: Graphic presentations of isotropic sysmtes,J. Comb. Theory Series B,45 (1988) 58–76.
A. Bouchet: Digraph decompositions and Eulerian systems,SIAM J. Alg. Discrete Math.,8 (1987) 323–337.
A. Bouchet Recognizing locally equivalent graphs, to appear inDiscrete Math.
A. Kotzig:Qualques remarques sur les transformations ϰ séminaire Paris, 1977.
D. G. Fon-Der-Flaass: On local complementations of graphs, in:Proc 7th Hungarian Colloquium of Combinatorics, July 1987, Colloquia Mathematica Societatis János Bolyai, pp. 257–266.
D. G. Fon-Der-Flaass: letter March 20, 1989.
Author information
Authors and Affiliations
Additional information
With partial support of P. R. C. Mathématiques et Informatique.
Rights and permissions
About this article
Cite this article
Bouchet, A. An efficient algorithm to recognize locally equivalent graphs. Combinatorica 11, 315–329 (1991). https://doi.org/10.1007/BF01275668
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01275668