A Decomposition Theorem for Binary Matroids with no Prism Minor

The prism graph is the dual of the complete graph on five vertices with an edge deleted, K5\e. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac’s infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connection of F7 with itself across a triangle with an element of the triangle deleted; it’s rank is bounded by 5; or it admits a non-minimal exact 3-separation induced by the 3separation in P9. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, F7 and PG(3, 2), respectively. We show that there is just one rank 5 extremal matroid in the class. It has 17 elements and is an extension of R10, the unique splitter for regular matroids. As a corollary, we obtain Mayhew and Royle’s result identifying the binary internally 4-connected matroids with no prism minor Mayhew and Royle (Siam J Discrete Math 26:755–767, 2012). The first author is partially supported by PSC-CUNY grant number 64181-00 42. The second author is partially supported by CNPq under Grant number 300242/2008-05. S. R. Kingan (B) Department of Mathematics Brooklyn College, City University of NewYork, Brooklyn, NY 11210, USA e-mail: [email protected] M. Lemos Departamento de Matematica, Universidade Federal de Pernambuco, Recife Pernambuco 50740-540, Brazil e-mail: [email protected]