Controlling representations of frame matroids

Report on joint work in progress with Dillon Mayhew, Victoria University of Wellington, New Zealand delivered at the www.canadam.ca/index.html">Canadian Discrete and Algorithmic Mathematics (CANDAM) Conference, 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.