Matroids with at least two regular elements

Sandra Kingan; Manoel Lemos

doi:10.1016/j.ejc.2011.12.006

paper

Matroids with at least two regular elements

arXiv:1103.2959 · doi:10.1016/j.ejc.2011.12.006

Abstract

For a matroid $M$ , an element $e$ such that both $M \ e$ and $M / e$ are regular is called a regular element of $M$ . We determine completely the structure of non-regular matroids with at least two regular elements. Besides four small size matroids, all 3-connected matroids in the class can be pieced together from $F_{7}$ or $S_{8}$ and a regular matroid using 3-sums. This result takes a step toward solving a problem posed by Paul Seymour: Find all 3-connected non-regular matroids with at least one regular element [5, 14.8.8].