# Pivot and Loop Complementation on Graphs and Set Systems

• Robert Brijder
• Hendrik Jan Hoogeboom
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6108)

## 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 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.

## References

1. 1.
Aigner, M., van der Holst, H.: Interlace polynomials. Linear Algebra and its Applications 377, 11–30 (2004)
2. 2.
Arratia, R., Bollobás, B., Sorkin, G.B.: The interlace polynomial: a new graph polynomial. In: SODA 2000: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 237–245. Society for Industrial and Applied Mathematics, Philadelphia (2000)Google Scholar
3. 3.
Arratia, R., Bollobás, B., Sorkin, G.B.: The interlace polynomial of a graph. Journal of Combinatorial Theory, Series B 92(2), 199–233 (2004)
4. 4.
Bouchet, A.: Representability of Δ-matroids. In: Proc. 6th Hungarian Colloquium of Combinatorics, Colloquia Mathematica Societatis János Bolyai, vol. 52, pp. 167–182. North-Holland, Amsterdam (1987)Google Scholar
5. 5.
Bouchet, A.: Graphic presentations of isotropic systems. Journal of Combinatorial Theory, Series B 45(1), 58–76 (1988)
6. 6.
Bouchet, A., Duchamp, A.: Representability of Δ-matroids over $$\emph{GF}(2)$$. Linear Algebra and its Applications 146, 67–78 (1991)
7. 7.
Brijder, R., Hoogeboom, H.J.: The group structure of pivot and loop complementation on graphs and set systems, arXiv:0909.4004 (2009)Google Scholar
8. 8.
Brijder, R., Hoogeboom, H.J.: Maximal pivots on graphs with an application to gene assembly, arXiv:0909.3789 (submitted 2009)Google Scholar
9. 9.
Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, San Diego (1992)
10. 10.
de Fraysseix, H.: Local complementation and interlacement graphs. Discrete Mathematics 33(1), 29–35 (1981)
11. 11.
Van den Nest, M., Dehaene, J., De Moor, B.: Graphical description of the action of local clifford transformations on graph states. Physical Review A 69(2), 022316 (2004)
12. 12.
Ehrenfeucht, A., Harju, T., Petre, I., Prescott, D.M., Rozenberg, G.: Computation in Living Cells – Gene Assembly in Ciliates. Springer, Heidelberg (2004)
13. 13.
Geelen, J.F.: A generalization of Tutte’s characterization of totally unimodular matrices. Journal of Combinatorial Theory, Series B 70, 101–117 (1997)
14. 14.
Godsil, C., Royle, G.: Algebraic Graph Theory. Springer, Heidelberg (2001)
15. 15.
Oum, S.: Rank-width and vertex-minors. Journal of Combinatorial Theory, Series B 95(1), 79–100 (2005)
16. 16.
Tsatsomeros, M.J.: Principal pivot transforms: properties and applications. Linear Algebra and its Applications 307(1-3), 151–165 (2000)
17. 17.
Tucker, A.W.: A combinatorial equivalence of matrices. In: Combinatorial Analysis, Proceedings of Symposia in Applied Mathematics, vol. X, pp. 129–140. American Mathematical Society, Providence (1960)Google Scholar