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arXiv · 1905.11363

Complementation, Local Complementation, and Switching in Binary Matroids

James Oxley·Jagdeep Singh
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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.

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James Oxley, Jagdeep Singh. 2019-05-27. Complementation, Local Complementation, and Switching in Binary Matroids. https://doi.org/10.1016/j.aam.2020.102041

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