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

An incidence Hopf Algebra of Convex Geometries

A lattice L is "meet-distributive" if for each element of L, the meets of the elements directly below it form a Boolean lattice. These objects are in bijection with "convex geometries", which are an abstract model of convexity. Do they give rise to an incidence Hopf algebra of convex geometries

The Forbidden Minor Characterization of Line-Search Antimatroids of Rooted Digraphs

An antimatroid is an accessible union-closed family of subsets of a finite set. A number of classes of antimatroids are closed under taking minors such as point-search antimatroids of rooted (di)graphs, line-search antimatroids of rooted (di)graphs, shelling antimatroids of rooted trees, shelling antimatroids of posets, etc. The forbidden minor characterizations are known for point-search antimatroids of rooted (di)graphs, shelling antimatroids of rooted trees and shelling antimatroids of posets. In this paper, we give the forbidden minor characterization of line-search antimatroids of rooted digraphs

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM