Facets of the p-cycle polytope

The purpose of this study is to provide a polyhedral analysis of the p-cycle polytope, which is the convex hull of the incidence vectors of all the p-cycles (simple directed cycles consisting of p arcs) of the complete directed graph Kn. We first determine the dimension of the p-cycle, polytope, characterize the bases of its equality set, and prove two lifting results. We then describe several classes of valid inequalities for the case 2<p<n, together with necessary and sufficient conditions for these inequalities to induce facets of the p-cycle polytope. We also briefly discuss the complexity of the associated separation problems. Finally, we investigate the relationship between the p-cycle polytope and related polytopes, including the p-circuit polytope. Since the undirected versions of symmetric inequalities which induce facets of the p-cycle polytope are facet-inducing for the p-circuit polytope, we obtain new classes of facet-inducing inequalities for the p-circuit polytope

Vertex-Facet Incidences of Unbounded Polyhedra

How much of the combinatorial structure of a pointed polyhedron is contained in its vertex-facet incidences? Not too much, in general, as we demonstrate by examples. However, one can tell from the incidence data whether the polyhedron is bounded. In the case of a polyhedron that is simple and "simplicial," i.e., a d-dimensional polyhedron that has d facets through each vertex and d vertices on each facet, we derive from the structure of the vertex-facet incidence matrix that the polyhedron is necessarily bounded. In particular, this yields a characterization of those polyhedra that have circulants as vertex-facet incidence matrices.Comment: LaTeX2e, 14 pages with 4 figure

Chiral extensions of chiral polytopes

Given a chiral d-polytope K with regular facets, we describe a construction for a chiral (d + 1)-polytope P with facets isomorphic to K. Furthermore, P is finite whenever K is finite. We provide explicit examples of chiral 4-polytopes constructed in this way from chiral toroidal maps.Comment: 21 pages. [v2] includes several minor revisions for clarit

Incompatible double posets and double order polytopes

In 1986 Stanley associated to a poset the order polytope. The close interplay between its combinatorial and geometric properties makes the order polytope an object of tremendous interest. Double posets were introduced in 2011 by Malvenuto and Reutenauer as a generalization of Stanleys labelled posets. A double poset is a finite set equipped with two partial orders. To a double poset Chappell, Friedl and Sanyal (2017) associated the double order polytope. They determined the combinatorial structure for the class of compatible double posets. In this paper we generalize their description to all double posets and we classify the 2-level double order polytopes.Comment: 11 pages, 3 figure