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

AbstractIn 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.Firstly we introduce the notion of closed set systems. This leads to a common generalization of rooted circuits both of matroids and convex geometries (antimatroids). We further study some properties of affine convex geometries and their broken circuit clutters

Decompositions of Simplicial Balls and Spheres With Knots Consisting of Few Edges

Constructibility is a condition on pure simplicial complexes that is weaker than shellability. In this paper we show that non-constructible triangulations of the d-dimensional sphere exist for every d 3. This answers a question of Danaraj & Klee [8]; it also strengthens a result of Lickorish [13] about non-shellable spheres. Furthermore, we provide a hierarchy of combinatorial decomposition properties that follow from the existence of a non-trivial knot with "few edges" in a 3-sphere or 3-ball, and a similar hierarchy for 3-balls with a knotted spanning arc that consists of "few edges.&quot