The group structure of pivot and loop complementation on graphs and set systems

Authors:
Robert Brijder;Hendrik Jan Hoogeboom
Affiliations:
Hasselt University and Transnational University of Limburg, Belgium;Leiden Institute of Advanced Computer Science, Leiden University, The Netherlands
Venue:
European Journal of Combinatorics
Year:
2011

Citing 8
Cited 1

Graphic presentations of isotropic systems

Journal of Combinatorial Theory Series A
A generalization of Tutte's characterization of totally unimodular matrices

Journal of Combinatorial Theory Series B - Special issue: dedicated to Professor W. T. Tutte on the occasion of his eightieth birthday
The interlace polynomial: a new graph polynomial

SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Multimatroids III. Tightness and fundamental graphs

European Journal of Combinatorics - Special issue on combinatorial geometries
Computation in Living Cells: Gene Assembly in Ciliates (Natural Computing Series)

Computation in Living Cells: Gene Assembly in Ciliates (Natural Computing Series)
Rank-width and vertex-minors

Journal of Combinatorial Theory Series B
Pivot and loop complementation on graphs and set systems

TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
Pivots, determinants, and perfect matchings of graphs

Theoretical Computer Science

Pivots, determinants, and perfect matchings of graphs

Theoretical Computer Science

Quantified Score

Hi-index	0.00

Visualization

Abstract

We study the interplay between the 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 S"3. 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.