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

作者：Brijder Robert^*; Hoogeboom Hendrik Jan

来源：European Journal of Combinatorics, 2011, 32(8): 1353-1367.

DOI：10.1016/j.ejc.2011.03.002

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.

• 出版日期2011-11