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Polyhedral aspects of partial orders and comparability graphs

par Fiorini, Samuel
Référence Electronic Notes in Discrete Mathematics, 2, page (171)
Publication Publié, 1999-04

Article révisé par les pairs

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Résumé :

Polyhedral combinatorics studies convex polytopes defined as convex hulls of certain combinatorial sets embedded in Rd. Important issues in the study of such polytopes are the determination of the facets, edges and automorphism group. We consider all partial orders on a given set of size n, represented in Rn(n-1) by means of their characteristic vectors. The associated convex hull is the partial order polytope. A remarkable feature is that axioms for partial orders define facets of the partial order polytope. Unfortunately, most facets have no such nice interpretation. Finding them all turns out to be quite hard. With the help of the computer program porta, we obtained all facets of the partial order polytope up to n = 4, and a partial list of facets for n = 5. This list contains a total of 43244 facets and we were able to prove it is incomplete thanks to a personal algorithm testing whether a list of facets of a polytope contains all facets. For all n, the automorphism group of the partial order polytope is entirely determined and some edges are characterized. The results on the edges give rise to a conjecture stating a strikingly simple criterion for adjacency. A simple, undirected graph is a comparability graph if its edges can be transitively oriented, thus producing a partial order. Using characteristic vectors in Rn(n-1)/2, we encode all comparability graphs on a given set of size n. The comparability graph polytope is the convex hull of these vectors. The latter appears as the image of the partial order polytope under a canonical projection which maps the characteristic vector of a partial order to the characteristic vector of its comparability graph. This projection can be exploited to infer new facets of the partial order polytope from facets of the comparability graph polytope. Some of these facets, that we call lifted odd cycle facets, are literal translations of the characterization of comparability graphs due to Ghouilà-Houri in 1962 (independently obtained by Gilmore and Hoffman in 1964). © 2000.