Matching 2-lattice polyhedra are a special class of lattice polyhedra that include network 3ow polyhedra, fractional matching polyhedra, matroid intersection polyhedra, etc. In this paper we develop a polynomial-time extreme point algorithm for #nding a maximum cardinality vector in a matching 2-lattice polyhedron. c © 2001 Elsevier Science B.V. All rights reserved.

In a 1979 paper by Coxeter and Huybers [4] , simple formulas were given to construct the snub cube and the snub dodecahedron from the octahedron and the icosahedron respectively , the vertices of the octahedron and icosahedron being the intersection points of certain pairs of edges of the two snub polyhedra . In a later series of papers , Hohenberg [6 , 7 , 8] studied the intersection points of...

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz’s theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric pro...

We show that up to unimodular equivalence there are only finitely many d-dimensional lattice polytopes without interior lattice points that do not admit a lattice projection onto a (d− 1)-dimensional lattice polytope without interior lattice points. This was conjectured by Treutlein. As an immediate corollary, we get a short proof of a recent result of Averkov, Wagner &Weismantel, namely the fi...

The Ehrhart polynomial of a lattice polygon P is completely determined by the pair (b(P ), i(P )) where b(P ) equals the number of lattice points on the boundary and i(P ) equals the number of interior lattice points. All possible pairs (b(P ), i(P )) are completely described by a theorem due to Scott. In this note, we describe the shape of the set of pairs (b(T ), i(T )) for lattice triangles ...

We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n , where n is the dimension. This improves on the previously best known bound n and complements a recent result of Braun [8] where it is shown that the norm of a root of a Ehrhart po...

In this paper we study the directed cut cone and polytope which are the positive hull and convex hull of all directed cut vectors of a complete directed graph, respectively. We present results on the polyhedral structure of these polyhedra. A relation between directed cut polyhedra and undirected cut polyhedra is established that provides families of facet defining inequalities for directed cut...

Local coordination and topology of ions determine several important properties of materials, including electronic structure, migration barrier, and diffusivity. In this thesis, we employ the Materials Project Database to investigate the coordination preferences of cations and topology of coordination polyhedra in oxides. We calculate the coordination environment preferences of several common ca...

Two-lattice polyhedra are a special class of lattice polyhedra that include network 4ow polyhedra, fractional matching polyhedra, matroid intersection polyhedra, the intersection of two polymatroids, etc. In this paper we show that the maximum sum of components of a vector in a 2-lattice polyhedron is equal to the minimum capacity of a cover for the polyhedron. For special classes of 2-lattice ...