Abstract
The operation of local complementation was introduced by A. Bouchet while studying Euler cycles in 4-regular graphs and graphs of chord intersections. This paper surveys some recent results on local complementations.
The notions of local complementation and local equivalence are generalized to the case of directed graphs. Several theorems analogous to those obtained earlier for simple graphs are proved for directed graphs. Among other results, the conjecture of A. Bouchet on simple graphs isomorphic to their local equivalents is proved.
This research was partially supported by the Russian Foundation for Fundamental Research (Grant 93–01–01486).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Kotzig (1968) Eulerian lines in finite 4-valent graphs and their transformations, in: Theory of Graphs, Academic Press, New York, pp. 219–230.
A. Bouchet (1994) Cycle graph obstructions, J. Combin. Theory Ser. В 60, No. 1, 107–144.
A. Bouchet (1990) k-transformations, local complementations, and switchings, in: Cycles and Rays, Kluwer Academic Publishers, Dordrecht, pp. 41–50.
A. Bouchet (1987) Reducing prime graphs and recognizing circle graphs, Combinatorica 7, No. 3, 243–254.
A. Bouchet (1988) Transforming trees by successive local complementations, J. Graph Theory 12, No. 2, 195–207.
A. Bouchet (1991) An efficient algorithm to recognize locally equivalent graphs, Combinatorica 11, No. 4, 315–329.
A. Bouchet (1987) Digraph decompositions and Eulerian systems, SIAM J. Algebraic Discrete Methods 8, No. 3, 323–337.
D. Fon-Der-Flaass (1988) On local complementations of graphs, in: Combinatorics, Coll. Math. Soc. J. Bolyai. Vol. 52, North-Holland, Amsterdam-Oxford-New York, pp. 257–266.
D. G. Fon-Der-Flaass (1989) Distance between locally equivalent graphs (in Russian), Metody Diskret. Analiz. 48, 85–94.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Kluwer Academic Publishers
About this chapter
Cite this chapter
Fon-Der-Flaass, D.G. (1996). Local Complementations of Simple and Directed Graphs. In: Korshunov, A.D. (eds) Discrete Analysis and Operations Research. Mathematics and Its Applications, vol 355. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1606-7_3
Download citation
DOI: https://doi.org/10.1007/978-94-009-1606-7_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7217-5
Online ISBN: 978-94-009-1606-7
eBook Packages: Springer Book Archive