ScienceDirect - Discrete Mathematics : Transitivity of local complementation and switching on graphs

Discrete Mathematics
Volume 278, Issues 1-3, 6 March 2004, Pages 45-60

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doi:10.1016/j.disc.2003.04.001

Transitivity of local complementation and switching on graphs

Andrzej Ehrenfeucht ^,^a, Tero Harju ^,^b and Grzegorz Rozenberg ^,^c

^a Department of Computer Science, University of Colorado at Boulder, Boulder, CO 80309-0347, USA ^b Department of Mathematics, University of Turku, FIN-20014, Turku, Finland ^c Leiden Institute of Advanced Computer Science, Leiden University, P.O. Box 9512, 2300 RA, Leiden, Netherlands

Received 30 January 2002;

Revised 28 April 2003;

accepted 26 May 2003.

Available online 1 March 2004.

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Abstract

The operations complementation C, local complementation λ_x, and switching σ_x for the vertices x of a finite undirected graph are considered. The operation λ_x complements the subgraph induced by the neighbourhood of x in the given graph, and the switching σ_x changes the neighbourhood of x to its complement vertex set. It is proved that the compositions δ_x=λ_xC (for vertices x set membership, variant D) generate a transitive group on the graphs with vertex set D, that is, for any two graphs g and h on D, there exists a composition α of operations δ_x such that h=α(g). It is also shown that the compositions τ_x=λ_xσ_x (for xD) generate a transitive group on the graphs.