Interval matroids and graphs

Abstract

A base of the cycle space of a binary matroid M on E is said to be convex if its elements can be totally ordered in such a way that for every e ε E the set of elements of the base containing e is an interval. We show that a binary matroid is cographic iff it has a convex base of cycles; equivalently, graphic matroids can be represented as “interval matroids” (matroids associated in a natural way to interval systems). As a consequence, we obtain characterizations of planar graphs and cubic cyclically-4-edge-connected planar graphs in terms of convex bases of cycles.