On Regular Matroids Without Certain Minors

Abstract

We extend several excluded-minor characterizations of graphs without certain minors to the class of regular matroids. By Seymour’s decomposition of the regular matroids, any regular matroid without a certain minor N can be constructed from N-free graphic matroids, cographic matroids, and \({R_10}\) (if \({R_10}\) is N-free) using 1-, 2-, and 3-sums. The converse, however, is not necessarily true. For example, the \({K_{3, 3}}\)-free graphs \({K_{5}\ e}\) and \({K_4}\) can be 3-summed over a certain triangle to form the graph \({K_{3, 3}}\). In this paper, we first prove a decomposition theorem of regular matroids without a vertically 4-connected minor with rank exceeding three. As a result, we extend to regular matroids some graph excluded minors results of Ding for the octahedron, Maharry for the cube, and Robertson for the Mobius ladder on eight vertices.