图和集合系统上的枢轴补和圈补的群结构
The Group Structure of Pivot and Loop Complementation on Graphs and Set Systems
摘要：
我们研究了图的主枢轴变换（枢轴）和循环补之间的相互作用。这是通过将循环互补（除了枢轴）推广到集合系统来实现的。我们证明了，当这些运算仅限于单顶点时，它们一起构成置换群S_3。这导致，例如，图上的枢轴序列和回路补的范式。的结果有后果的操作简单图的局部互补和边缘互补：一个经典的结果，涉及局部和边缘互补的替代证明，并获得了简单图的局部互补序列的效果的特点。
英文摘要：
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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