27 research outputs found
Two Results in Drawing Graphs on Surfaces
In this work we present results on crossing-critical graphs drawn on non-planar surfaces and results on edge-hamiltonicity of graphs on the Klein bottle. We first give an infinite family of graphs that are 2-crossing-critical on the projective plane. Using this result, we construct 2-crossing-critical graphs for each non-orientable surface. Next, we use 2-amalgamations to construct 2-crossing-critical graphs for each orientable surface other than the sphere. Finally, we contribute to the pursuit of characterizing 4-connected graphs that embed on the Klein bottle and fail to be edge-hamiltonian. We show that known 4-connected counterexamples to edge-hamiltonicity on the Klein bottle are hamiltonian and their structure allows restoration of edge-hamiltonicity with only a small change
Delaunay Tessellations and Voronoi Diagrams in CGAL
The Cgal library provides a rich variety of Voronoi diagrams and Delaunay triangulations. This variety covers several aspects: generators, dimensions and metrics, which we describe in Section 2. One aim of this paper is to present the main paradigms used in CGAL: Generic programming, separation between predicates/constructions and combinatorics, and exact geometric computation (not to be confused with exact arithmetic!). The first two paradigms translate into software design choices, described in Section 4, while the last covers both robustness and efficiency issues, respectively described in Sec- tion 6 and 7. Other important aspects of the Cgal library are the interface issues, be they for traversing a tessellation, or for interoperability with other libraries or languages, see Section 5. We present in Section 8 some tessellations at work in the context of surface reconstruction and mesh generation. Section 9 is devoted to some on-going and future work on periodic triangulations (triangulations in periodic spaces), and on high-quality mesh generation with optimized tessellations. Section 10 provides typical numbers in terms of efficiency and scalability for constructing tessellations, and lists the remaining weaknesses. We conclude by listing some of our directions for the future
Toric Topology
Toric topology emerged in the end of the 1990s on the borders of equivariant
topology, algebraic and symplectic geometry, combinatorics and commutative
algebra. It has quickly grown up into a very active area with many
interdisciplinary links and applications, and continues to attract experts from
different fields.
The key players in toric topology are moment-angle manifolds, a family of
manifolds with torus actions defined in combinatorial terms. Their construction
links to combinatorial geometry and algebraic geometry of toric varieties via
the related notion of a quasitoric manifold. Discovery of remarkable geometric
structures on moment-angle manifolds led to seminal connections with the
classical and modern areas of symplectic, Lagrangian and non-Kaehler complex
geometry. A related categorical construction of moment-angle complexes and
their generalisations, polyhedral products, provides a universal framework for
many fundamental constructions of homotopical topology. The study of polyhedral
products is now evolving into a separate area of homotopy theory, with strong
links to other areas of toric topology. A new perspective on torus action has
also contributed to the development of classical areas of algebraic topology,
such as complex cobordism.
The book contains lots of open problems and is addressed to experts
interested in new ideas linking all the subjects involved, as well as to
graduate students and young researchers ready to enter into a beautiful new
area.Comment: Preliminary version. Contains 9 chapters, 5 appendices, bibliography,
index. 495 pages. Comments and suggestions are very welcom
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
SMARANDACHE MULTI-SPACE THEORY, Second Edition
We are used to the idea that our space has three dimensions: length, breadth and height with time providing the fourth dimension of spacetime by Einstein. In the string or superstring theories, we encounter 10 dimensions. However, we do not even know what the right degree of freedom is, as Witten said. Today, we have known two heartening notions for sciences. One is the Smarandache multi-space came into being by purely logic. Another is the mathematical combinatorics motivated by a combinatorial speculation, i.e., a mathematical science can be reconstructed from or made by combinatorialization. Both of them contribute sciences for consistency of research with that human progress in 21st century