Some extremal connectivity results for matroids

Abstract

Let n be an integer exceeding one and M be a matroid having at least n + 2 elements. In this paper, we prove that every n-element subset X of E(M) is in an (n + 1)-element circuit if and only if (i) for every such subset, M X is disconnected, and (ii) for every subset Y with at most n elements, M Y is connected. Various extensions and consequences of this result are also derived including characterizations in terms of connectivity of the 4-point line and of Murty\u27s Sylvester matroids. The former is a result of Seymour. © 1991