Cubic Time Recognition of Cocircuit Graphs of Uniform Oriented Matroids

We present an algorithm which takes a graph as input and decides in cubic time if the graph is the cocircuit graph of a uniform oriented matroid. In the affirmative case the algorithm returns the set of signed cocircuits of the oriented matroid. This improves an algorithm proposed by Babson, Finschi and Fukuda. Moreover we strengthen a result of Montellano-Ballesteros and Strausz about crabbed connectivity of cocircuit graphs of uniform oriented matroids.Comment: 9 page

Erdős–Szekeres “happy end”-type theorems for separoïds

AbstractIn 1935 Pál Erdős and György Szekeres proved that, roughly speaking, any configuration of n points in general position in the plane have logn points in convex position — which are the vertices of a convex polygon. Later, in 1983, Bernhard Korte and László Lovász generalised this result in a purely combinatorial context; the context of greedoids. In this note we give one step further to generalise this last result for arbitrary dimensions, but in the context of separoids; thus, via the geometric representation theorem for separoids, this can be applied to families of convex bodies. Also, it is observed that the existence of some homomorphisms of separoids implies the existence of not-too-small polytopal subfamilies — where each body is separated from its relative complement. Finally, by means of a probabilistic argument, it is settled, basically, that for all d>2, asymptotically almost all “simple” families of n “d-separated” convex bodies contains a polytopal subfamily of order lognd+1

A generalisation of Tverberg’s theorem

Non UBCUnreviewedAuthor affiliation: Universidad Nacional Autonoma de MexicoFacult

Universality of separoids

summary:A separoid is a symmetric relation \dagger \subset {2^S\atopwithdelims ()2} defined on disjoint pairs of subsets of a given set $S$ such that it is closed as a filter in the canonical partial order induced by the inclusion (i.e., $A\dagger B\preceq A^{\prime }\dagger B^{\prime }\iff A\subseteq A^{\prime }$ and $B\subseteq B^{\prime }$). We introduce the notion of homomorphism as a map which preserve the so-called “minimal Radon partitions” and show that separoids, endowed with these maps, admits an embedding from the category of all finite graphs. This proves that separoids constitute a countable universal partial order. Furthermore, by embedding also all hypergraphs (all set systems) into such a category, we prove a “stronger” universality property. We further study some structural aspects of the category of separoids. We completely solve the density problem for (all) separoids as well as for separoids of points. We also generalise the classic Radon’s theorem in a categorical setting as well as Hedetniemi’s product conjecture (which can be proved for oriented matroids)