Weakly Modular Graphs and Nonpositive Curvature

This article investigates structural, geometrical, and topological characterizations and properties of weakly modular graphs and of cell complexes derived from them. The unifying themes of our investigation are various “nonpositive curvature" and “local-to-global” properties and characterizatio...

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Hlavní autoři: Chalopin, Jérémie, Chepoi, Victor, Hirai, Hiroshi, Osajda, Damian
Médium: E-kniha Kniha
Jazyk:angličtina
Vydáno: Providence, Rhode Island American Mathematical Society 2020
Vydání:1
Edice:Memoirs of the American Mathematical Society
Témata:
Curvature
Distance geometry
Graph theory
combinatorial optimization
lattice
nonpositive curvature
CAT space
incidence geometry
Weakly modular graph
group action
building
ISBN:9781470443627, 1470443627
ISSN:0065-9266, 1947-6221
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Obsah:
  • Introduction -- Preliminaries -- Local-to-Global Characterization -- Pre-Median Graphs -- Dual Polar Graphs -- Sweakly Modular Graphs -- Orthoscheme Complexes of Modular Lattices and Semilattices -- Orthoscheme Complexes of Swm-Graphs -- Metric Properties of Weakly Modular Graphs
  • Cover -- Title page -- Chapter 1. Introduction -- 1.1. Avant-propos -- 1.2. Motivation -- 1.3. Main results -- Acknowledgments -- Chapter 2. Preliminaries -- 2.1. Basic notions -- 2.2. Further notions -- Chapter 3. Local-to-Global Characterization -- 3.1. Main results -- 3.2. Weakly modular graphs: proofs -- 3.3. Helly and clique-Helly graphs: proofs -- 3.4. A note on meshed graphs -- Chapter 4. Pre-Median Graphs -- 4.1. Main results -- 4.2. Prime pre-median graphs -- 4.3. Examples -- 4.4. ₁-Weakly modular graphs -- 4.5. ( ) is contractible -- Chapter 5. Dual Polar Graphs -- 5.1. Main results -- 5.2. Proof of Theorem 5.2 -- 5.3. Proof of Theorems 5.3 and 5.4 -- Chapter 6. Sweakly Modular Graphs -- 6.1. Main results -- 6.2. A lattice-theoretical characterization of swm-graphs -- 6.3. Boolean pairs and Boolean-gated sets -- 6.4. Barycentric graph * -- 6.5. Thickening ^{Δ} -- 6.6. Δ-gates and geodesic extension property -- 6.7. Normal Boolean-gated paths -- 6.8. Euclidean buildings of type C_{ } -- 6.9. Application I: Biautomaticity of swm-groups -- 6.10. Application II: 2-approximation to 0-extension problem on swm-graphs -- Chapter 7. Orthoscheme Complexes of Modular Lattices and Semilattices -- 7.1. Main results -- 7.2. Boolean and complemented modular lattices -- 7.3. Distributive and modular lattices -- 7.4. Proof of Theorem 7.2 -- 7.5. Proof of Propositions 7.5 and 7.6 -- 7.6. Proof of Proposition 7.4 -- Chapter 8. Orthoscheme Complexes of Swm-Graphs -- 8.1. Main results -- 8.2. Examples -- 8.3. Preliminary results -- 8.4. ₁-metrization -- 8.5. _{∞}-metrization -- 8.6. ₂-metrization -- 8.7. ( ) is contractible -- Chapter 9. Metric Properties of Weakly Modular Graphs -- 9.1. Quadratic isoperimetric inequality -- 9.2. Hyperbolicity -- 9.3. BFS gives a distance-preserving ordering -- 9.4. Weakly modular complex -- Bibliography
  • Back Cover