One characterization of binary matroids is that the symmetric difference of every pair of intersecting circuits is a disjoint union of circuits. This paper considers circuit-difference matroids, that is, those matroids in which the symmetric difference of every pair of intersecting circuits is a single circuit. Our main result shows that a connected regular matroid is circuit-difference if and only if it contains no pair of skew circuits. Using a result of Pfeil, this enables us to explicitly determine all regular circuit-difference matroids. The class of circuit-difference matroids is not closed under minors, but it is closed under series minors. We characterize the infinitely many excluded series minors for the class.
Publication Source (Journal or Book title)
Electronic Journal of Combinatorics
Drummond, G., Oxley, J., Grace, K., & Fife, T. (2020). Circuit-difference matroids. Electronic Journal of Combinatorics, 27 (3), 1-11. https://doi.org/10.37236/9314