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Weakly Modular Graphs and Nonpositive Curvature
About this Title
Jérémie Chalopin, Victor Chepoi, Hiroshi Hirai and Damian Osajda
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 268, Number 1309
ISBNs: 978-1-4704-4362-7 (print); 978-1-4704-6349-6 (online)
DOI: https://doi.org/10.1090/memo/1309
Published electronically: March 1, 2021
Keywords: Weakly modular graph,
nonpositive curvature,
CAT(0) space,
lattice,
building,
incidence geometry,
group action,
combinatorial optimization
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Local-to-Global Characterization
- 4. Pre-Median Graphs
- 5. Dual Polar Graphs
- 6. Sweakly Modular Graphs
- 7. Orthoscheme Complexes of Modular Lattices and Semilattices
- 8. Orthoscheme Complexes of Swm-Graphs
- 9. Metric Properties of Weakly Modular Graphs
Abstract
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 characterizations of weakly modular graphs and their subclasses. Weakly modular graphs have been introduced as a far-reaching common generalization of median graphs (and more generally, of modular and orientable modular graphs), Helly graphs, bridged graphs, and dual polar graphs occurring under different disguises ($1$–skeletons, collinearity graphs, covering graphs, domains, etc.) in several seemingly-unrelated fields of mathematics:
- Peter Abramenko and Kenneth S. Brown, Buildings, Graduate Texts in Mathematics, vol. 248, Springer, New York, 2008. Theory and applications. MR 2439729
- Ian Agol, The virtual Haken conjecture, Doc. Math. 18 (2013), 1045–1087. With an appendix by Agol, Daniel Groves, and Jason Manning. MR 3104553, DOI 10.1016/j.procs.2013.05.269
- Federico Ardila, Megan Owen, and Seth Sullivant, Geodesics in $\rm CAT(0)$ cubical complexes, Adv. in Appl. Math. 48 (2012), no. 1, 142–163. MR 2845512, DOI 10.1016/j.aam.2011.06.004
- N. Aronszajn and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405–439. MR 84762
- H.-J. Bandelt, Retracts of hypercubes, J. Graph Theory 8 (1984), no. 4, 501–510. MR 766499, DOI 10.1002/jgt.3190080407
- H.-J. Bandelt, Hereditary modular graphs, Combinatorica 8 (1988), no. 2, 149–157. MR 963122, DOI 10.1007/BF02122796
- Hans-Jürgen Bandelt and Victor Chepoi, A Helly theorem in weakly modular space, Discrete Math. 160 (1996), no. 1-3, 25–39. MR 1417558, DOI 10.1016/0012-365X(95)00217-K
- Hans-Jürgen Bandelt and Victor Chepoi, Decomposition and $l_1$-embedding of weakly median graphs, European J. Combin. 21 (2000), no. 6, 701–714. Discrete metric spaces (Marseille, 1998). MR 1791200, DOI 10.1006/eujc.1999.0377
- Hans-Jürgen Bandelt and Victor Chepoi, Metric graph theory and geometry: a survey, Surveys on discrete and computational geometry, Contemp. Math., vol. 453, Amer. Math. Soc., Providence, RI, 2008, pp. 49–86. MR 2405677, DOI 10.1090/conm/453/08795
- Hans-Jürgen Bandelt and Henry Martyn Mulder, Pseudomodular graphs, Discrete Math. 62 (1986), no. 3, 245–260. MR 866940, DOI 10.1016/0012-365X(86)90212-8
- Hans-Jürgen Bandelt, Henry Martyn Mulder, and Elke Wilkeit, Quasi-median graphs and algebras, J. Graph Theory 18 (1994), no. 7, 681–703. MR 1297190, DOI 10.1002/jgt.3190180705
- Hans-Jürgen Bandelt and Erwin Pesch, Dismantling absolute retracts of reflexive graphs, European J. Combin. 10 (1989), no. 3, 211–220. MR 1029165, DOI 10.1016/S0195-6698(89)80053-8
- Hans-Jürgen Bandelt and Erich Prisner, Clique graphs and Helly graphs, J. Combin. Theory Ser. B 51 (1991), no. 1, 34–45. MR 1088625, DOI 10.1016/0095-8956(91)90004-4
- H.-J. Bandelt and M. van de Vel, Superextensions and the depth of median graphs, J. Combin. Theory Ser. A 57 (1991), no. 2, 187–202. MR 1111556, DOI 10.1016/0097-3165(91)90044-H
- H.-J. Bandelt, M. van de Vel, and E. Verheul, Modular interval spaces, Math. Nachr. 163 (1993), 177–201. MR 1235066, DOI 10.1002/mana.19931630117
- Jean-Pierre Barthélémy and Julien Constantin, Median graphs, parallelism and posets, Discrete Math. 111 (1993), no. 1-3, 49–63. Graph theory and combinatorics (Marseille-Luminy, 1990). MR 1210081, DOI 10.1016/0012-365X(93)90140-O
- Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
- A. Björner, Topological methods, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 1819–1872. MR 1373690
- Tom Brady and Jon McCammond, Braids, posets and orthoschemes, Algebr. Geom. Topol. 10 (2010), no. 4, 2277–2314. MR 2745672, DOI 10.2140/agt.2010.10.2277
- B. Brešar, J. Chalopin, V. Chepoi, T. Gologranc, and D. Osajda, Bucolic complexes, Adv. Math. 243 (2013), 127–167. MR 3062742, DOI 10.1016/j.aim.2013.04.009
- Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486
- A. E. Brouwer and A. M. Cohen, Local recognition of Tits geometries of classical type, Geom. Dedicata 20 (1986), no. 2, 181–199. MR 833846, DOI 10.1007/BF00164399
- A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 18, Springer-Verlag, Berlin, 1989. MR 1002568
- A. E. Brouwer and H. A. Wilbrink, The structure of near polygons with quads, Geom. Dedicata 14 (1983), no. 2, 145–176. MR 708631, DOI 10.1007/BF00181622
- Peter J. Cameron, Dual polar spaces, Geom. Dedicata 12 (1982), no. 1, 75–85. MR 645040, DOI 10.1007/BF00147332
- Jérémie Chalopin, Victor Chepoi, and Damian Osajda, On two conjectures of Maurer concerning basis graphs of matroids, J. Combin. Theory Ser. B 114 (2015), 1–32. MR 3354288, DOI 10.1016/j.jctb.2015.03.004
- Jérémie Chalopin, Victor Chepoi, Panos Papasoglu, and Timothée Pecatte, Cop and robber game and hyperbolicity, SIAM J. Discrete Math. 28 (2014), no. 4, 1987–2007. MR 3284564, DOI 10.1137/130941328
- Marc Chastand, Fiber-complemented graphs. I. Structure and invariant subgraphs, Discrete Math. 226 (2001), no. 1-3, 107–141. MR 1801065, DOI 10.1016/S0012-365X(00)00183-7
- Marc Chastand, Fiber-complemented graphs. II. Retractions and endomorphisms, Discrete Math. 268 (2003), no. 1-3, 81–101. MR 1982390, DOI 10.1016/S0012-365X(02)00682-9
- Indira Chatterji, Cornelia Druţu, and Frédéric Haglund, Kazhdan and Haagerup properties from the median viewpoint, Adv. Math. 225 (2010), no. 2, 882–921. MR 2671183, DOI 10.1016/j.aim.2010.03.012
- V. D. Chepoĭ, Classification of graphs by means of metric triangles, Metody Diskret. Analiz. 49 (1989), 75–93, 96 (Russian). MR 1114014
- Victor Chepoi, A multifacility location problem on median spaces, Discrete Appl. Math. 64 (1996), no. 1, 1–29. MR 1367177, DOI 10.1016/0166-218X(95)00115-8
- Victor Chepoi, Bridged graphs are cop-win graphs: an algorithmic proof, J. Combin. Theory Ser. B 69 (1997), no. 1, 97–100. MR 1426753, DOI 10.1006/jctb.1996.1726
- Victor Chepoi, On distance-preserving and domination elimination orderings, SIAM J. Discrete Math. 11 (1998), no. 3, 414–436. MR 1628110, DOI 10.1137/S0895480195291230
- Victor Chepoi, Graphs of some $\textrm {CAT}(0)$ complexes, Adv. in Appl. Math. 24 (2000), no. 2, 125–179. MR 1748966, DOI 10.1006/aama.1999.0677
- Victor Chepoi, Basis graphs of even delta-matroids, J. Combin. Theory Ser. B 97 (2007), no. 2, 175–192. MR 2290319, DOI 10.1016/j.jctb.2006.05.003
- Victor Chepoi, Nice labeling problem for event structures: a counterexample, SIAM J. Comput. 41 (2012), no. 4, 715–727. MR 2974750, DOI 10.1137/110837760
- Victor Chepoi, Feodor F. Dragan, Bertrand Estellon, Michel Habib, and Yann Vaxès, Diameters, centers, and approximating trees of $\delta$-hyperbolic geodesic spaces and graphs, Computational geometry (SCG’08), ACM, New York, 2008, pp. 59–68. MR 2504271, DOI 10.1145/1377676.1377687
- Victor Chepoi and Mark F. Hagen, On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes, J. Combin. Theory Ser. B 103 (2013), no. 4, 428–467. MR 3071375, DOI 10.1016/j.jctb.2013.04.003
- Victor Chepoi and Daniela Maftuleac, Shortest path problem in rectangular complexes of global nonpositive curvature, Comput. Geom. 46 (2013), no. 1, 51–64. MR 2949610, DOI 10.1016/j.comgeo.2012.04.002
- Victor Chepoi and Damian Osajda, Dismantlability of weakly systolic complexes and applications, Trans. Amer. Math. Soc. 367 (2015), no. 2, 1247–1272. MR 3280043, DOI 10.1090/S0002-9947-2014-06137-0
- E. Dahlhaus, D. S. Johnson, C. H. Papadimitriou, P. D. Seymour, and M. Yannakakis, The complexity of multiterminal cuts, SIAM J. Comput. 23 (1994), no. 4, 864–894. MR 1283579, DOI 10.1137/S0097539792225297
- Michael W. Davis, The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series, vol. 32, Princeton University Press, Princeton, NJ, 2008. MR 2360474
- F. Dragan, HT-graphs: centers, connected $r$-domination and Steiner trees, Comput. Sci. J. Moldova 1 (1993), no. 2, 64–83. MR 1317867
- Michel Marie Deza and Monique Laurent, Geometry of cuts and metrics, Algorithms and Combinatorics, vol. 15, Springer-Verlag, Berlin, 1997. MR 1460488
- Andreas W. M. Dress and Rudolf Scharlau, Gated sets in metric spaces, Aequationes Math. 34 (1987), no. 1, 112–120. MR 915878, DOI 10.1007/BF01840131
- Jack Edmonds, Submodular functions, matroids, and certain polyhedra, Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969) Gordon and Breach, New York, 1970, pp. 69–87. MR 0270945
- David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. MR 1161694
- Martin Farber and Robert E. Jamison, On local convexity in graphs, Discrete Math. 66 (1987), no. 3, 231–247. MR 900046, DOI 10.1016/0012-365X(87)90099-9
- I. M. Gel′fand, R. M. Goresky, R. D. MacPherson, and V. V. Serganova, Combinatorial geometries, convex polyhedra, and Schubert cells, Adv. in Math. 63 (1987), no. 3, 301–316. MR 877789, DOI 10.1016/0001-8708(87)90059-4
- R. L. Graham and P. M. Winkler, On isometric embeddings of graphs, Trans. Amer. Math. Soc. 288 (1985), no. 2, 527–536. MR 776391, DOI 10.1090/S0002-9947-1985-0776391-5
- George Grätzer, Lattice theory: foundation, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2768581
- M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
- Thomas Haettel, Dawid Kielak, and Petra Schwer, The 6-strand braid group is $\textrm {CAT}(0)$, Geom. Dedicata 182 (2016), 263–286. MR 3500387, DOI 10.1007/s10711-015-0138-9
- Frédéric Haglund, Complexes simpliciaux hyperboliques de grande dimension, Prepublication Orsay 71 (2003), preprint, available at http://www.math.u-psud.fr/~biblio/ppo/2003/fic/ppo_2003_71.pdf.
- Frédéric Haglund and Frédéric Paulin, Simplicité de groupes d’automorphismes d’espaces à courbure négative, The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, 1998, pp. 181–248 (French, with English and French summaries). MR 1668359, DOI 10.2140/gtm.1998.1.181
- Frédéric Haglund and Daniel T. Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008), no. 5, 1551–1620. MR 2377497, DOI 10.1007/s00039-007-0629-4
- Wolfgang Haken, Connections between topological and group theoretical decision problems, Word problems: decision problems and the Burnside problem in group theory (Conf., Univ. California, Irvine, Calif., 1969; dedicated to Hanna Neumann), North-Holland, Amsterdam, 1973, pp. 427–441. Studies in Logic and the Foundations of Math., Vol. 71. MR 0397736
- Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
- Pavol Hell and Ivan Rival, Absolute retracts and varieties of reflexive graphs, Canad. J. Math. 39 (1987), no. 3, 544–567. MR 905743, DOI 10.4153/CJM-1987-025-1
- Hiroshi Hirai, Folder complexes and multiflow combinatorial dualities, SIAM J. Discrete Math. 25 (2011), no. 3, 1119–1143. MR 2825329, DOI 10.1137/090767054
- Hiroshi Hirai, Discrete convexity and polynomial solvability in minimum 0-extension problems, Math. Program. 155 (2016), no. 1-2, Ser. A, 1–55. MR 3439796, DOI 10.1007/s10107-014-0824-7
- J. R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964), 65–76. MR 182949, DOI 10.1007/BF02566944
- John R. Isbell, Median algebra, Trans. Amer. Math. Soc. 260 (1980), no. 2, 319–362. MR 574784, DOI 10.1090/S0002-9947-1980-0574784-8
- Tadeusz Januszkiewicz and Jacek Świa̧tkowski, Simplicial nonpositive curvature, Publ. Math. Inst. Hautes Études Sci. 104 (2006), 1–85. MR 2264834, DOI 10.1007/s10240-006-0038-5
- Howard Karloff, Subhash Khot, Aranyak Mehta, and Yuval Rabani, On earthmover distance, metric labeling, and 0-extension, SIAM J. Comput. 39 (2009), no. 2, 371–387. MR 2520313, DOI 10.1137/070685671
- Alexander V. Karzanov, Minimum $0$-extensions of graph metrics, European J. Combin. 19 (1998), no. 1, 71–101. MR 1600283, DOI 10.1006/eujc.1997.0154
- Alexander V. Karzanov, Metrics with finite sets of primitive extensions, Ann. Comb. 2 (1998), no. 3, 211–241. MR 1681515, DOI 10.1007/BF01608533
- Alexander V. Karzanov, One more well-solved case of the multifacility location problem, Discrete Optim. 1 (2004), no. 1, 51–66. MR 2098080, DOI 10.1016/j.disopt.2004.04.001
- Jon Kleinberg and Éva Tardos, Approximation algorithms for classification problems with pairwise relationships: metric labeling and Markov random fields, J. ACM 49 (2002), no. 5, 616–639. MR 2145145, DOI 10.1145/585265.585268
- J. H. Koolen, V. Moulton, and D. Stevanović, The structure of spherical graphs, European J. Combin. 25 (2004), no. 2, 299–310. MR 2070550, DOI 10.1016/S0195-6698(03)00116-1
- Francisco Larrión, Miguel A. Pizaña, and Rafael Villarroel-Flores, The fundamental group of clique-Helly graphs, Mat. Contemporânea 39 (2010), 5-7.
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. MR 1812024
- Jie Hua Mai and Yun Tang, An injective metrization for collapsible polyhedra, Proc. Amer. Math. Soc. 88 (1983), no. 2, 333–337. MR 695270, DOI 10.1090/S0002-9939-1983-0695270-9
- Rajsekar Manokaran, Joseph Naor, Prasad Raghavendra, and Roy Schwartz, SDP gaps and UGC hardness for multiway cut, 0-extension, and metric labeling [extended abstract], STOC’08, ACM, New York, 2008, pp. 11–20. MR 2582655, DOI 10.1145/1374376.1374379
- Jiří Matoušek, Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, 2002. MR 1899299
- Stephen B. Maurer, Matroid basis graphs. I, J. Combinatorial Theory Ser. B 14 (1973), 216–240. MR 317971, DOI 10.1016/0095-8956(73)90005-1
- Stephen B. Maurer, Matroid basis graphs. II, J. Combinatorial Theory Ser. B 15 (1973), 121–145. MR 369112, DOI 10.1016/0095-8956(73)90013-0
- Shahar Mozes, Products of trees, lattices and simple groups, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 571–582. MR 1648106
- G. A. Niblo and L. D. Reeves, The geometry of cube complexes and the complexity of their fundamental groups, Topology 37 (1998), no. 3, 621–633. MR 1604899, DOI 10.1016/S0040-9383(97)00018-9
- Gennady A. Noskov, Combing Euclidean buildings, Geom. Topol. 4 (2000), 85–116. MR 1735633, DOI 10.2140/gt.2000.4.85
- Damian Osajda, A combinatorial non-positive curvature I: weak systolicity (2013), preprint, available at arXiv:1305.4661.
- Damian Osajda, A construction of hyperbolic Coxeter groups, Comment. Math. Helv. 88 (2013), no. 2, 353–367. MR 3048190, DOI 10.4171/CMH/288
- Damian Osajda, Combinatorial negative curvature and triangulations of three-manifolds, Indiana Univ. Math. J. 64 (2015), no. 3, 943–956. MR 3361292, DOI 10.1512/iumj.2015.64.5568
- P. Papasoglu, Strongly geodesically automatic groups are hyperbolic, Invent. Math. 121 (1995), no. 2, 323–334. MR 1346209, DOI 10.1007/BF01884301
- Norbert Polat, On infinite bridged graphs and strongly dismantlable graphs, Discrete Math. 211 (2000), no. 1-3, 153–166. MR 1735348, DOI 10.1016/S0012-365X(99)00142-9
- Norbert Polat, Convexity and fixed-point properties in Helly graphs, Discrete Math. 229 (2001), no. 1-3, 197–211. Combinatorics, graph theory, algorithms and applications. MR 1815607, DOI 10.1016/S0012-365X(00)00210-7
- Erich Prisner, Convergence of iterated clique graphs, Discrete Math. 103 (1992), no. 2, 199–207. MR 1171317, DOI 10.1016/0012-365X(92)90270-P
- Piotr Przytycki and Jennifer Schultens, Contractibility of the Kakimizu complex and symmetric Seifert surfaces, Trans. Amer. Math. Soc. 364 (2012), no. 3, 1489–1508. MR 2869183, DOI 10.1090/S0002-9947-2011-05465-6
- Michah Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. (3) 71 (1995), no. 3, 585–617. MR 1347406, DOI 10.1112/plms/s3-71.3.585
- Thomas J. Schaefer, The complexity of satisfiability problems, Conference Record of the Tenth Annual ACM Symposium on Theory of Computing (San Diego, Calif., 1978) ACM, New York, 1978, pp. 216–226. MR 521057
- S. V. Shpectorov, On scale embeddings of graphs into hypercubes, European J. Combin. 14 (1993), no. 2, 117–130. MR 1206617, DOI 10.1006/eujc.1993.1016
- Ernest E. Shult, Points and lines, Universitext, Springer, Heidelberg, 2011. Characterizing the classical geometries. MR 2761484
- Ernest Shult and Arthur Yanushka, Near $n$-gons and line systems, Geom. Dedicata 9 (1980), no. 1, 1–72. MR 566437, DOI 10.1007/BF00156473
- V. P. Soltan and V. D. Chepoĭ, Conditions for invariance of set diameters under $d$-convexification in a graph, Kibernetika (Kiev) 6 (1983), 14–18 (Russian, with English summary); English transl., Cybernetics 19 (1983), no. 6, 750–756 (1984). MR 765117, DOI 10.1007/BF01068561
- Jayme L. Szwarcfiter, Recognizing clique-Helly graphs, Ars Combin. 45 (1997), 29–32. MR 1447758
- Jacek Świȧtkowski, Regular path systems and (bi)automatic groups, Geom. Dedicata 118 (2006), 23–48. MR 2239447, DOI 10.1007/s10711-005-9003-6
- Jacques Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974. MR 0470099
- Johannes Ueberberg, Foundations of incidence geometry, Springer Monographs in Mathematics, Springer, Heidelberg, 2011. Projective and polar spaces. MR 3025282
- M. L. J. van de Vel, Theory of convex structures, North-Holland Mathematical Library, vol. 50, North-Holland Publishing Co., Amsterdam, 1993. MR 1234493
- M. van de Vel, Collapsible polyhedra and median spaces, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2811–2818. MR 1452832, DOI 10.1090/S0002-9939-98-04413-X
- Vijay V. Vazirani, Approximation algorithms, Springer-Verlag, Berlin, 2001. MR 1851303
- Glynn Winskel and Mogens Nielsen, Models for concurrency, Handbook of logic in computer science, Vol. 4, Handb. Log. Comput. Sci., vol. 4, Oxford Univ. Press, New York, 1995, pp. 1–148. MR 1365754
- Daniel T. Wise, Sixtolic complexes and their fundamental groups (2003), unpublished manuscript.
- D. T. Wise, Cubulating small cancellation groups, Geom. Funct. Anal. 14 (2004), no. 1, 150–214. MR 2053602, DOI 10.1007/s00039-004-0454-y
- Daniel T. Wise, The structure of groups with quasiconvex hierarchy, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 2017, to appear.
Metric graph theory
Geometric group theory
Incidence geometries and buildings
Theoretical computer science and combinatorial optimization
We give a local-to-global characterization of weakly modular graphs and their subclasses in terms of simple connectedness of associated triangle-square complexes and specific local combinatorial conditions. In particular, we revisit characterizations of dual polar graphs by Cameron and by Brouwer-Cohen. We also show that (disk-)Helly graphs are precisely the clique-Helly graphs with simply connected clique complexes. With $l_1$–embeddable weakly modular and sweakly modular graphs we associate high-dimensional cell complexes, having several strong topological and geometrical properties (contractibility and the CAT(0) property). Their cells have a specific structure: they are basis polyhedra of even $\triangle$–matroids in the first case and orthoscheme complexes of gated dual polar subgraphs in the second case. We resolve some open problems concerning subclasses of weakly modular graphs: we prove a Brady-McCammond conjecture about CAT(0) metric on the orthoscheme complexes of modular lattices; we answer Chastand’s question about prime graphs for pre-median graphs. We also explore negative curvature for weakly modular graphs.