The Group Structure of Pivot and Loop Complementation on Graphs and Set Systems

TL;DR

This paper explores the algebraic structure of pivot and loop complementation operations on graphs and set systems, revealing their group properties and implications for graph transformations like local and edge complementation.

Contribution

It generalizes loop complementation to set systems and characterizes the group structure of pivot and loop operations, providing a normal form and new insights into graph transformations.

Findings

Operations form the permutation group S_3 when restricted to single vertices

Provides a normal form for sequences of pivots and loop complementations

Offers an alternative proof and characterization of local and edge complementation effects

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.