The max-flow min-cut property of two-dimensional affine convex geometries

摘要

In a matroid, (X,e) is a rooted circuit if X is a set not containing element e and X{e} is a circuit. We call X a broken circuit of e. A broken circuit clutter is the collection of broken circuits of a fixed element. Seymour [The matroids with the max-flow min-cut property, J. Combinatorial Theory B 23 (1977) 189–222] proved that a broken circuit clutter of a binary matroid has the max-flow min-cut property if and only if it does not contain a minor isomorphic to Q6. We shall present an analogue of this result in affine convex geometries. Precisely, we shall show that a broken circuit clutter of an element e in a convex geometry arising from two-dimensional point configuration has the max-flow min-cut property if and only if the configuration has no subset forming a ‘Pentagon’ configuration with center e.