Frame Matroids and Biased Graphs

Abstract

A frame matroid is any submatroid of a matroid in which each point belongs to a line spanned by a fixed basis. A biased graph is a graph with certain polygons called balanced, no theta graph containing exactly two balanced polygons. We prove that certain matroids, called bias matroids, of biased graphs are identical to the finitary frame matroids. As an application we deduce two simple characterizations of frame matroids and some facts about planar forbidden minors for bias matroids.