We describe how some simple properties of discrete one-forms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated "spring-embedding" theorem for planar graphs, which is widely used for parameterizing meshes with the topology of a disk as a planar embedding with a convex boundary. Our second result generalizes the first, dealing with the case where the mesh contains multiple boundaries, which are free to be non-convex in the embedding. We characterize when it is still possible to achieve an embedding, despite these boundaries being non-convex. The third result is an analogous embedding theorem for meshes with genus 1 (topologically equivalent to the torus). Applications of these results to the parameterization of meshes with disk and toroidal topologies are demonstrated. Extensions to higher genus meshes are discussed.

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Abstract. In a paper presented at a 1996 conference, Holt and Klee introduced a set of necessary conditions for an orientation of the graph of a d-polytope to be induced by a realization into R n and linear functional on that space. In general, it is an open question to decide whether for a polytope P every orientation of its graph satisfying these conditions can in fact be realized in this fashion; two natural families of polytopes to consider are cubes and crosspolytopes. For cubes, we show that, as n grows, the percentage of n-cube Holt-Klee orientations which can be realized goes asymptotically to 0. For crosspolytopes, we give a stronger set of conditions which are both necessary and sufficient for an orientation to be in this class; as a corollary, we prove that all shellings of cubes are line shellings. 1.

In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound theorem for the face numbers of convex polytopes and its many extensions is the second topic. Next are general properties of subsets of the vertices of the discrete n-dimensional cube and some relations with questions of extremal and probabilistic combinatorics. Our fourth topic is tree enumeration and random spanning trees, and finally, some combinatorial and geometrical aspects of the simplex method for linear programming are considered.

INTRODUCTION The k-dimensional skeleton of a d-polytope P is the set of all faces of the polytope of dimension at most k. The 1-skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs--- subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t

. We investigate the vertex-connectivity of the graph of f-monotone paths on a d-polytope P with respect to a generic functional f . The third author has conjectured that this graph is always (d \Gamma 1)-connected. We resolve this conjecture positively for simple polytopes and show that the graph is 2connected for any d-polytope with d 3. However, we disprove the conjecture in general by exhibiting counterexamples for each d 4 in which the graph has a vertex of degree two. We also re-examine the Baues problem for cellular strings on polytopes, solved by Billera, Kapranov and Sturmfels. Our analysis shows that their positive result is a direct consequence of shellability of polytopes and is therefore less related to convexity than is at first apparent. 1. Introduction Let P be a d-dimensional polytope in R d and f be a linear functional on R d which is generic with respect to P , in the sense that f is nonconstant on every edge of P . An f-monotone path fl on P is a se...

The Monotone Upper Bound Problem asks for the maximal number M(d, n) ofvertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d, n) ≤ Mubt(d, n) provided by McMullen’s (1970) Upper Bound Theorem is tight, where Mubt(d, n) is the number of vertices of a dual-to-cyclic d-polytope with n facets. It was recently shown that the upper bound M(d, n) ≤ Mubt(d, n) holdswith equality for small dimensions (d ≤ 4: Pfeifle, 2003) and for small corank (n ≤ d +2: Gärtner et al., 2001). Here we prove that it is not tight in general: In dimension d =6apolytopewithn =9facetscanhaveMubt(6, 9) = 30 vertices, but not more than 27 ≤ M(6, 9) ≤ 29 vertices can lie on a strictly-increasing edge-path. The proof involves classification results about neighborly polytopes, Kalai’s (1988) concept of abstract objective functions, the Holt-Klee conditions (1998), explicit enumeration, Welzl’s (2001) extended Gale diagrams, randomized generation of instances, as well as non-realizability proofs via a version of the Farkas lemma.

We investigate the worst-case behavior of the simplex algorithm on linear programs with 3 variables, that is, on 3-dimensional simple polytopes. Among the pivot rules that we consider, the “random edge” rule yields the best asymptotic behavior as well as the most complicated analysis. All other rules turn out to be much easier to study, but also produce worse results: Most of them show essentially worst-possible behavior; this includes both Kalai’s “random-facet” rule, which is known to be subexponential without dimension restriction, as well as Zadeh’s deterministic history-dependent rule, for which no non-polynomial instances in general dimensions have been found so far.

We investigate the combinatorial structure of linear programs on simple d-polytopes with d + 2 facets. These can be encoded by admissible grid orientations. Admissible grid orientations are also obtained through orientation properties of a planar point configuration or by the dual line arrangement. The point configuration and the polytope corresponding to the same grid are related through an extended Gale transform. The class of admissible grid orientations is shown to contain non-realizable examples, i.e., there are admissible grid orientations which cannot be obtained from a polytope or a point configuration. It is shown, however, that every admissible grid orientation is induced by an arrangement of pseudolines. This later result is used to prove several nontrivial facts about admissible grid orientations.

The possible pivot operations of the simplex method to solve a linear program can be represented as a directed graph defined on the skeleton of the feasible region P. We consider the case that P is bounded, i.e., a convex polytope. The directed graph is called an LP digraph. LP digraphs are known to satisfy the following three properties: acyclicity, unique sink orientation(USO), and the Holt-Klee property. The three properties are not generally sufficient for a directed graph on the skeleton of P to be an LP digraph. In this paper, we first survey some previous results on LP digraphs, showing relationships among the three properties. Then we introduce a new necessary property for a directed graph on the skeleton of P to be an LP digraph, called the shelling property. We analyze the relationships between the shelling property and the three existing properties, showing that it is stronger than a combination of acyclicity and USO for non-simple polytopes in dimension at least four. In all other cases it is equivalent to the intersection of these two properties. 1