Complementation, Local Complementation, and Switching in Binary Matroids
Abstract
In 2004, Ehrenfeucht, Harju, and Rozenberg showed that any graph on a vertex set can be obtained from a complete graph on 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 under the operations of complementation and switching. Moreover, we show that not all binary matroids of rank at most can be obtained from a projective geometry of rank 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.
Cite
@article{arxiv.1905.11363,
title = {Complementation, Local Complementation, and Switching in Binary Matroids},
author = {James Oxley and Jagdeep Singh},
journal= {arXiv preprint arXiv:1905.11363},
year = {2020}
}
Comments
Fixed an error in the proof of Theorem 5.3. Adv. in Appl. Math. (2020)