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The Group Structure of Pivot and Loop Complementation on Graphs and Set Systems
- Source :
- European Journal of Combinatorics, v. 32, 1353-1367, 2011
- Publication Year :
- 2009
-
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.<br />Comment: 21 pages, 7 figures, significant additions w.r.t. v3 are Thm 7 and Remark 25
Details
- Database :
- arXiv
- Journal :
- European Journal of Combinatorics, v. 32, 1353-1367, 2011
- Publication Type :
- Report
- Accession number :
- edsarx.0909.4004
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.ejc.2011.03.002