Pivot and loop complementation on graphs and set systems

Authors:
Robert Brijder;Hendrik Jan Hoogeboom
Affiliations:
Leiden Institute of Advanced Computer Science, Leiden University, The Netherlands;Leiden Institute of Advanced Computer Science, Leiden University, The Netherlands
Venue:
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
Year:
2010

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Abstract

We study the interplay between principal pivot transform (pivot) and loop complementation for graphs This is done by generalizing loop complementation (in addition to pivot) to set systems We show that the operations together, when restricted to single vertices, form the permutation group S3 This leads, e.g., to a normal form for sequences of pivots and loop complementation on graphs The results have consequences for the operations of local complementation and edge complementation on simple graphs: an alternative proof of a classic result involving local and edge complementation is obtained, and the effect of sequences of local complementations on simple graphs is characterized.