Pivot and Loop Complementation on Graphs and Set Systems

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


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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Robert Brijder
    • 1
  • Hendrik Jan Hoogeboom
    • 1
  1. 1.Leiden Institute of Advanced Computer ScienceLeiden UniversityThe Netherlands

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