On the cogirth of binary matroids
Document Type
Article
Publication Date
6-1-2023
Abstract
The cogirth, g⁎(M), of a matroid M is the size of a smallest cocircuit of M. Finding the cogirth of a graphic matroid can be done in polynomial time, but Vardy showed in 1997 that it is NP-hard to find the cogirth of a binary matroid. In this paper, we show that [Formula presented] when M is binary, unless M simplifies to a projective geometry. We also show that, when equality holds, M simplifies to a Bose-Burton geometry, that is, a matroid of the form PG(r−1,2)−PG(k−1,2). These results extend to matroids representable over arbitrary finite fields.
Publication Source (Journal or Book title)
Advances in Applied Mathematics
Recommended Citation
Crenshaw, C., & Oxley, J. (2023). On the cogirth of binary matroids. Advances in Applied Mathematics, 147 https://doi.org/10.1016/j.aam.2023.102515