# Complementation, Local Complementation, and Switching in Binary Matroids

## TL;DR

This paper generalizes graph operations like complementation and switching to binary matroids, characterizes their effects on binary projective geometries, and introduces a new operation to generate all binary matroids.

## Contribution

It extends graph operations to binary matroids, characterizes their effects, and introduces a new operation to generate all binary matroids from projective geometries.

## Key findings

- Not all binary matroids of a given rank can be obtained from a projective geometry using three operations.
- A fourth operation is introduced that allows generating all binary matroids.
- The paper provides a characterization of binary matroids obtainable from projective geometries.

## Abstract

In 2004, Ehrenfeucht, Harju, and Rozenberg showed that any graph on a vertex set $V$ can be obtained from a complete graph on $V$ via a sequence of the operations of complementation, switching edges and non-edges at a vertex, and local complementation. The last operation involves taking the complement in the neighbourhood of a vertex. In this paper, we consider natural generalizations of these operations for binary matroids and explore their behaviour. We characterize all binary matroids obtainable from the binary projective geometry of rank $r$ under the operations of complementation and switching. Moreover, we show that not all binary matroids of rank at most $r$ can be obtained from a projective geometry of rank $r$ via a sequence of the three generalized operations. We introduce a fourth operation and show that, with this additional operation, we are able to obtain all binary matroids.