Report on joint work in progress with Dillon Mayhew, Victoria University of Wellington, New Zealand delivered at the <a href="https://www.canadam.ca/index.html">Canadian Discrete and Algorithmic Mathematics (CANDAM) Conference</a>, Winnipeg, Manitoba (June 5-8, 2023).
A matroid is an abstract object that generalises both graphs and vector spaces. Matroids are used to model many types of optimisation problems; often modelling a problem using a matroid can lead to an efficient algorithm for finding optimal solutions. Frame matroids are an important type of matroid, and frame matroids can be represented by biased graphs. Unfortunately, understanding all the biased graph representations of a given frame matroids is difficult, and little is known. We present a theorem which provides a rough biased graphical structure for representations of frame matroids that are sufficiently large, and discuss implications for understanding those substructures that cannot occur in any frame matroid.
We conjecture that the class of frame matroids can be characterized by a sentence in the monadic second-order logic of matroids, and we prove that there is such a characterization for the class of bicircular matroids. The proof does not depend on an excluded-minor characterization.
