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Die Graphentheorie gewinnt einen Teil ihres Reizes aus ihrer Anschaulichkeit, also aus Bezügen zu Visualisierung und Geometrie. Die Graphikfähigkeit moderner Computer hat neue Anstöße zur Untersuchung dieser Bezüge geliefert, insbesondere Geometrische Graphen sind zu einem aktuellenForschungsgegenstand geworden. Arrangements von Geraden und Punkten sind die Objekte einer Fülle von oftmals schwierigen Problemen und überraschenden Lösungen. Dieses Buch, in englischer Sprache, stellt faszinierende und teils sehr aktuelle Themen aus dem Grenzbereich von Graphentheorie, Geometrie und Kombinatorik vor.
List of contents
1 Geometric Graphs: Turán Problems.- 1.1 What is a Geometric Graph?.- 1.2 Fundamental Concepts in Graph Theory.- 1.3 Planar Graphs.- 1.4 Outerplanar Graphs and Convex Geometric Graphs.- 1.5 Geometric Graphs without (k + 1)-Pairwise Disjoint Edges.- 1.6 Geometric Graphs without Parallel Edges.- 1.7 Notes and References.- 2 Schnyder Woods or How to Draw a Planar Graph?.- 2.1 Schnyder Labelings and Woods.- 2.2 Regions and Coordinates.- 2.3 Geodesic Embeddings of Planar Graphs.- 2.4 Dual Schnyder Woods.- 2.5 Order Dimension of 3-Polytopes.- 2.6 Existence of Schnyder Labelings.- 2.7 Notes and References.- 3 Topological Graphs: Crossing Lemma and Applications.- 3.1 Crossing Numbers.- 3.2 Bounds for the Crossing Number.- 3.3 Improving the Crossing Constant.- 3.4 Crossing Numbers and Incidence Problems.- 3.5 Notes and References.- 4 k-Sets and k-Facets.- 4.1 k-Sets in the Plane.- 4.2 Beyond the Plane.- 4.3 The Rectilinear Crossing Number of Kn.- 4.4 Notes and References.- 5 Combinatorial Problems for Sets of Points and Lines.- 5.1 Arrangements, Planes, Duality.- 5.2 Sylvester's Problem.- 5.3 How many Lines are Spanned by n Points?.- 5.4 Triangles in Arrangements.- 5.5 Notes and References.- 6 Combinatorial Representations of Arrangements of Pseudolines.- 6.1 Marked Arrangements and Sweeps.- 6.2 Allowable Sequences and Wiring Diagrams.- 6.3 Local Sequences.- 6.4 Zonotopal Tilings.- 6.5 Triangle Signs.- 6.6 Signotopes and their Orders.- 6.7 Notes and References.- 7 Triangulations and Flips.- 7.1 Degrees in the Flip-Graph.- 7.2 Delaunay Triangulations.- 7.3 Regular Triangulations and Secondary Polytopes.- 7.4 The Associahedron and Catalan families.- 7.5 The Diameter of Gn and Hyperbolic Geometry.- 7.6 Notes and References.- 8 Rigidity and Pseudotriangulations.- 8.1 Rigidity,Motion and Stress.- 8.2 Pseudotriangles and Pseudotriangulations.- 8.3 Expansive Motions.- 8.4 The Polyhedron of of Pointed Pseudotriangulations.- 8.5 Expansive Motions and Straightening Linkages.- 8.6 Notes and References.
About the author
Prof. Dr. Stefan Felsner, Institut für Mathematik, Technische Universität Berlin, Germany.
Summary
Among the intuitively appealing aspects of graph theory is its close connection to drawings and geometry. The development of computer technology has become a source of motivation to reconsider these connections, in particular geometric graphs are emerging as a new subfield of graph theory. Arrangements of points and lines are the objects for many challenging problems and surprising solutions in combinatorial geometry. The book is a collection of beautiful and mostly very recent results from the intersection of geometry, graph theory and combinatorics.
Foreword
Graph theory, geometry and combinatorics brought together to generate a wealth of beauty in ideas
Additional text
"The book is written in a pleasant and clear style, with generous pictures and lucid explanations. [...] I recommend this splendid litte book für PhD students and researchers who work or wish to work in discrete geometry".
Combinatorics, Probability and Computing (Cambridge University Press), 15/2006
"[The author] has contributed an introduction to this fascinating and mathematically challenging - yet intuitively accessible - field."
Monatshefte für Mathematik, 02/2006
Report
"The book is written in a pleasant and clear style, with generous pictures and lucid explanations. [...] I recommend this splendid litte book für PhD students and researchers who work or wish to work in discrete geometry".
Combinatorics, Probability and Computing (Cambridge University Press), 15/2006
"[The author] has contributed an introduction to this fascinating and mathematically challenging - yet intuitively accessible - field."
Monatshefte für Mathematik, 02/2006
