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**1 - 6**of**6**### Cubic Time Recognition of Cocircuit Graphs of Uniform Oriented Matroids

, 2010

"... 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 ..."

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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. 1

### Realisation of separoids and a Tverberg-type problem

"... A separoid is a symmetric relation † ⊂ ( 2 S) 2 defined on pairs of disjoint subsets which is closed as a filter in the natural partial order (i.e., A † B ≼ C † D ⇐ ⇒ A ⊆ C and B ⊆ D). We discus the Geometric Representation Theorem for separoids: every separoid (S, †) can be represented by a famil ..."

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A separoid is a symmetric relation † ⊂ ( 2 S) 2 defined on pairs of disjoint subsets which is closed as a filter in the natural partial order (i.e., A † B ≼ C † D ⇐ ⇒ A ⊆ C and B ⊆ D). We discus the Geometric Representation Theorem for separoids: every separoid (S, †) can be represented by a family of (convex) polytopes, and their Radon partitions, in the Euclidean space of dimension |S | − 1. Furthermore, we introduce a new kind of separoids’ morphisms — called chromomorphisms — which allow us to study Tverberg’s generalisation (1966) of Radon’s theorem (1921) in the context of convex sets. In particular the following Tverberg-type theorem is proved: Let S be a separoid of order |S | = (k − 1)(d(S) + 1) + 1, where d(S) denotes the (combinatorial) dimension of S. If there exists a monomorphism S → P into a separoid of points in general position in IE d, then there exists a chromomorphism S − → Kk onto the complete separoid of order k. This theorem is, in a sense, dual to the Hadwiger-type theorem proved by

### UNIVERSALITY OF SEPAROIDS

"... Abstract. A separoid is a symmetric relation † ⊂ `2 S´ defined on disjoint 2 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 † B � A ′ † B ′ ⇐ ⇒ A ⊆ A ′ and B ⊆ B ′). We introduce the notion of homomorphism as a ..."

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Abstract. A separoid is a symmetric relation † ⊂ `2 S´ defined on disjoint 2 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 † B � A ′ † B ′ ⇐ ⇒ A ⊆ A ′ and B ⊆ B ′). 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).