Graphs of polyhedra; polyhedra as graphs

AbstractRelations between graph theory and polyhedra are presented in two contexts. In the first, the symbiotic dependence between 3-connected planar graphs and convex polyhedra is described in detail. In the second, a theory of nonconvex polyhedra is based on a graph-theoretic foundation. This approach eliminates the vagueness and inconsistency that pervade much of the literature dealing with polyhedra more general than the convex ones

Review of Matrices and Graphs in Geometry. Encyclopedia of Mathematics and its Applications, vol. 139 by Miroslav Fiedler, Cambridge University Press (2011). viii + 197 pp., ISBN: 978-0-521-46193-1 Miroslav Fiedler, Matrices and Graphs in Geometry. Encyclopedia of Mathematics and its Applications, vol. 139, Cambridge University Press, 2011., viii + 197 pp., ISBN 978-0-521-46193-1.

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An enumeration of simplicial 4-polytopes with 8 vertices

AbstractAn enumeration of all the different combinatorial types of 4-dimensional simplicial convex polytopes with 8 vertices is given. It corrects an earlier enumeration attempt by M. Brückner, and leads to a simple example of a diagram which is not a Schlegel diagram

Isohedra with dart-shaped faces

AbstractA polyhedron in E3 is said to be isohedral (or an isohedron) if its faces are equivalent under the action of its group of symmetries. We use Möbius nets of the three reflection groups of the five Platonic solids to construct isohedra whose faces are dart-shaped, and whose edges lie in planes of reflective symmetry of the polyhedron. This technique for constructing isohedra has only recently been used; it yields many new results in addition to those described in this paper. In the final section we also describe some other isohedra with dart-shaped faces

Shortness exponents of families of graphs

AbstractKnown estimates of the maximal length of simple circuits in certain 3-connected planar graphs are surveyed and improved in several directions

Deletion constructions of symmetric 4-configurations. Part I.

By deletion constructions we mean several methods of generation of new geometric configurations by the judicious deletion of certain points and lines, and introduction of other lines or points. A number of such procedures have recently been developed in a systematic way. We present here one family of such constructions, and will describe other families in the following parts

Designing Fair Ranking Schemes

Items from a database are often ranked based on a combination of multiple criteria. A user may have the flexibility to accept combinations that weigh these criteria differently, within limits. On the other hand, this choice of weights can greatly affect the fairness of the produced ranking. In this paper, we develop a system that helps users choose criterion weights that lead to greater fairness. We consider ranking functions that compute the score of each item as a weighted sum of (numeric) attribute values, and then sort items on their score. Each ranking function can be expressed as a vector of weights, or as a point in a multi-dimensional space. For a broad range of fairness criteria, we show how to efficiently identify regions in this space that satisfy these criteria. Using this identification method, our system is able to tell users whether their proposed ranking function satisfies the desired fairness criteria and, if it does not, to suggest the smallest modification that does. We develop user-controllable approximation that and indexing techniques that are applied during preprocessing, and support sub-second response times during the online phase. Our extensive experiments on real datasets demonstrate that our methods are able to find solutions that satisfy fairness criteria effectively and efficiently