Internally 4-Connected Binary Matroids Without a Prism+e Minor

Abstract

We give a complete characterization of the internally 4-connected binary matroids with no minor that is isomorphic to the cycle matroid of the prism+e graph. This characterization generalizes a result first proved by Mayhew and Royle [6] for binary matroids, and generalizes results of Dirac [2] and Lovasz [5] for graphs. Then we use the main result to show that a 3-connected binary matroid with no minor that is isomorphic to the cycle matroid of the prism+e graph is either one of a list of 132 matroids or can be constructed from six specific matroids using the matroid operations of parallel extension and 3-sum. Both MACEK and SAGE Matroid Package are used to assist our computations.