Skip to main content Accessibility help
×
Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-30T16:19:49.132Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  14 March 2024

Guillermo Pineda Villavicencio
Affiliation:
Deakin University, Victoria
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Polytopes and Graphs , pp. 428 - 444
Publisher: Cambridge University Press
Print publication year: 2024

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abbott, T. G. 2008. Generalizations of Kempe’s universality theorem. M.Phil. thesis, Massachusetts Institute of Technology. (353)Google Scholar
Adiprasito, K. 2019. FAQ on the g-theorem and the hard Lefschetz theorem for face rings. Rend. Mat. Appl., VII. Ser., 40(2), 97111. (393)Google Scholar
Adiprasito, K., Papadakis, S. A., and Petrotou, V. 2021. Anisotropy, biased pairings, and the Lefschetz property for pseudomanifolds and cycles. arXiv:2101.07245. (393)Google Scholar
Adiprasito, K. A., and Benedetti, B. 2014. The Hirsch conjecture holds for normal flag complexes. Math. Oper. Res., 39(4), 13401348. (335, 338)CrossRefGoogle Scholar
Adiprasito, K. A., and Sanyal, R. 2016. Relative Stanley-Reisner theory and upper bound theorems for Minkowski sums. Publ. Math., Inst. Hautes Étud. Sci., 124, 99163. (277, 304)CrossRefGoogle Scholar
Adler, I., and Dantzig, G. 1974. Maximum diameter of abstract polytopes. Math. Program. Study, 1, 2040. (335, 336)CrossRefGoogle Scholar
Alon, N., and Kalai, G. 1985. A simple proof of the upper bound theorem. Eur. J. Comb., 6(3), 211214. (393)CrossRefGoogle Scholar
Altshuler, A., and Shemer, I. 1984. Construction theorems for polytopes. Isr. J. Math., 47(2–3), 99110. (105, 153)CrossRefGoogle Scholar
Andreev, E. M. 1970a. On convex polyhedra in Lobachevskij spaces. Math. USSR, Sb., 10, 413440. (212)CrossRefGoogle Scholar
Andreev, E. M. 1970b. On convex polyhedra of finite volume in Lobachevskii space. Math. USSR, Sb., 12, 255259. (212)CrossRefGoogle Scholar
Ardila, F., and Develin, M. 2009. Tropical hyperplane arrangements and oriented matroids. Math. Z., 262(4), 795816. (110)CrossRefGoogle Scholar
Asimow, L., and Roth, B. 1978. The rigidity of graphs. Trans. Amer. Math. Soc., 245, 279289. (352)CrossRefGoogle Scholar
Asimow, L., and Roth, B. 1979. The rigidity of graphs. II. J. Math. Anal. Appl., 68(1), 171190. (360)CrossRefGoogle Scholar
Athanasiadis, C. A. 2009. On the graph connectivity of skeleta of convex polytopes. Discrete Comput. Geom., 42(2), 155165. (231, 252)CrossRefGoogle Scholar
Athanasiadis, C. A. 2011. Some combinatorial properties of flag simplicial pseudomanifolds and spheres. Ark. Mat., 49(1), 1729. (221)CrossRefGoogle Scholar
Avis, D., and Moriyama, S. 2009. On combinatorial properties of linear program digraphs. Pages 113 of: Polyhedral computation. CRM Proc. Lecture Notes, vol. 48. Providence, RI: AMS. (176)CrossRefGoogle Scholar
Babson, E., Finschi, L., and Fukuda, K. 2001. Cocircuit graphs and efficient orientation reconstruction in oriented matroids. Eur. J. Comb., 22(5), 587600. (271)CrossRefGoogle Scholar
Babson, E. K., Billera, L. J., and Chan, C. S. 1997. Neighborly cubical spheres and a cubical lower bound conjecture. Isr. J. Math., 102, 297315. (263, 391, 399)CrossRefGoogle Scholar
Bajmóczy, E. G., and Bárány, I. 1979. On a common generalization of Borsuk’s and Radon’s theorem. Acta Math. Acad. Sci. Hung., 34(3), 347350. (43)CrossRefGoogle Scholar
Balinski, M. L. 1961. On the graph structure of convex polyhedra in n-space. Pac. J. Math., 11, 431434. (161, 231, 232, 254)CrossRefGoogle Scholar
Bárány, I. 1982. A generalization of Carathéodory’s theorem. Discrete Math., 40(2–3), 141152. (42)CrossRefGoogle Scholar
Bárány, I. 2021. Combinatorial convexity. Providence, RI: AMS. (42, 43)CrossRefGoogle Scholar
Bárány, I., and Rote, G. 2006. Strictly convex drawings of planar graphs. Doc. Math., 11, 369391. (198)CrossRefGoogle Scholar
Barnette, D. W. 1969. A simple 4-dimensional nonfacet. Isr. J. Math., 7, 1620. (339, 389)CrossRefGoogle Scholar
Barnette, D. W. 1971. The minimum number of vertices of a simple polytope. Isr. J. Math., 10, 121125. (236, 350, 371, 372, 374, 378, 392)CrossRefGoogle Scholar
Barnette, D. W. 1973a. Generating planar 4-connected graphs. Isr. J. Math., 14, 113. (221)CrossRefGoogle Scholar
Barnette, D. W. 1973b. Graph theorems for manifolds. Isr. J. Math., 16, 6272. (254)CrossRefGoogle Scholar
Barnette, D. W. 1973c. A proof of the lower bound conjecture for convex polytopes. Pac. J. Math., 46, 349354. (236, 340, 350, 371, 372, 374, 378, 392, 393)CrossRefGoogle Scholar
Barnette, D. W. 1974a. The projection of the f-vectors of 4-polytopes onto the (E, S)-plane. Discrete Math., 10, 201216. (341, 343)CrossRefGoogle Scholar
Barnette, D. W. 1974b. An upper bound for the diameter of a polytope. Discrete Math., 10, 913. (319, 323, 327, 339)CrossRefGoogle Scholar
Barnette, D. W. 1982. Decompositions of homology manifolds and their graphs. Isr. J. Math., 41, 203212. (254)CrossRefGoogle Scholar
Barnette, D. W. 1987a. 5-connected 3-polytopes are refinements of octahedra. J. Comb. Theory, Ser. B, 42(2), 250254. (221, 228, 395)CrossRefGoogle Scholar
Barnette, D. W. 1987b. Two “simple” 3-spheres. Discrete Math., 67, 9799. (212, 272, 397)CrossRefGoogle Scholar
Barnette, D. W., and Grünbaum, B. 1969. On Steinitz’s theorem concerning convex 3-polytopes and on some properties of planar graphs. Pages 2740 of: The many facets of graph theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968). Berlin: Springer-Verlag. (212)CrossRefGoogle Scholar
Barnette, D. W., and Grünbaum, B. 1970. Preassigning the shape of a face. Pac. J. Math., 32, 299306. (212)CrossRefGoogle Scholar
Barnette, D. W., and Reay, J. R. 1973. Projections of f-vectors of four-polytopes. J. Comb. Theory, Ser. A, 15, 200209. (341, 342)CrossRefGoogle Scholar
Batagelj, V. 1989. An improved inductive definition of two restricted classes of triangulations of the plane. Pages 1118 of: Combinatorics and graph theory, vol. 25. Warsaw: Banach Cent. Publ. (221, 228, 395)Google Scholar
Bayer, M. 1987. The extended f-vectors of 4-polytopes. J. Comb. Theory, Ser. A, 44, 141151. (386, 387, 393)CrossRefGoogle Scholar
Bayer, M. M. 2018. Graphs, skeleta and reconstruction of polytopes. Acta Math. Hung., 155, 6173. (272)Google Scholar
Bayer, M. M., and Billera, L. J. 1984. Counting faces and chains in polytopes and posets. Pages 207252 of: Green, C. (ed), Combinatorics and algebra (Proc. Conf., Boulder/Colo. 1983), Contemp. Math., vol. 34. Providence, RI: Amer. Math. Soc. (393)Google Scholar
Bayer, M. M., and Billera, L. J. 1985. Generalized Dehn–Sommerville relations for polytopes, spheres and Eulerian partially ordered sets. Invent. Math., 79, 143157. (340, 381, 390)CrossRefGoogle Scholar
Beineke, L. W. 1968. On derived graphs and digraphs. Beitr. Graphentheorie, Int. Kolloquium Manebach (DDR) 1967, 17–23 (1968). (426)Google Scholar
Beineke, L. W. 1970. Characterizations of derived graphs. J. Comb. Theory, 9, 129135. (426)CrossRefGoogle Scholar
Berger, M. 2009. Geometry I. Universitext. Berlin: Springer-Verlag. Translated from the 1977 French original by M. Cole and S. Levy. (16, 42, 153)Google Scholar
Billera, L. J., and Lee, C. W. 1981. A proof of the sufficiency of McMullen’s conditions for f-vectors of simplicial convex polytopes. J. Comb. Theory, Ser. B, 31(3), 237255. (372, 378)CrossRefGoogle Scholar
Björner, A., and Vorwerk, K. 2015. On the connectivity of manifold graphs. Proc. Am. Math. Soc., 143(10), 41234132. (254)CrossRefGoogle Scholar
Björner, A., Edelman, P. H., and Ziegler, G. M. 1990. Hyperplane arrangements with a lattice of regions. Discrete Comput. Geom., 5(3), 263288. (270)CrossRefGoogle Scholar
Blind, R., and Blind, G. 1994. Gaps in the numbers of vertices of cubical polytopes, I. Discrete Comput. Geom., 11(3), 351356. (184, 229)CrossRefGoogle Scholar
Blind, G., and Blind, R. 1998. The almost simple cubical polytopes. Discrete Math., 184(1–3), 2548. (93)CrossRefGoogle Scholar
Blind, G., and Blind, R. 1999. Shellings and the lower bound theorem. Discrete Comput. Geom., 21(4), 519526. (374, 393)CrossRefGoogle Scholar
Blind, G., and Blind, R. 2003. On a class of equifacetted polytopes. Pages 6978 of: Discrete geometry. Monogr. Textbooks Pure Appl. Math., vol. 253. New York: Dekker. (389, 392, 399)Google Scholar
Blind, R., and Mani-Levitska, P. 1987. Puzzles and polytope isomorphisms. Aequationes Math., 34(2–3), 287297. (256)CrossRefGoogle Scholar
Bollobás, B., and Thomason, A. 1996. Highly linked graphs. Combinatorica, 16(3), 313320. (255)Google Scholar
Bondy, J. A., and Murty, U. S. R. 2008. Graph theory. Graduate Texts in Mathematics, vol. 244. New York: Springer. (xi, 164, 407, 408, 409, 410, 417, 422, 427)CrossRefGoogle Scholar
Bonichon, N., Felsner, S., and Mosbah, M. 2007. Convex drawings of 3-connected plane graphs. Algorithmica, 47(4), 399420. (198)CrossRefGoogle Scholar
Bonifas, N., Di Summa, M., Eisenbrand, F., Hähnle, N., and Niemeier, M. 2014. On sub-determinants and the diameter of polyhedra. Discrete Comput. Geom., 52(1), 102115. (335)CrossRefGoogle Scholar
Brehm, U., and Sarkaria, K. S. 1992. Linear vs piecewise linear embeddability of simplicial complexes. 92/52. Max-Planck-Institut für Mathematik. (225)Google Scholar
Bremner, D., and Schewe, L. 2011. Edge-graph diameter bounds for convex polytopes with few facets. Exp. Math., 20(3), 229237. (335)CrossRefGoogle Scholar
Bricard, R. 1897. Mémoire sur la théorie de l’octaèdre articulé. J. Math. Pures Appl., 3, 113148. (366)Google Scholar
Brightwell, G. R., and Scheinerman, E. R. 1993. Representations of planar graphs. SIAM J. Discrete Math., 6(2), 214229. (214)CrossRefGoogle Scholar
Brinkmann, G., Greenberg, S., Greenhill, C., McKay, B.D, Thomas, R., and Wollan, P. 2005. Generation of simple quadrangulations of the sphere. Discrete Math., 305(1), 3354. (221, 228, 230, 395)CrossRefGoogle Scholar
Brøndsted, A. 1982. A dual proof of the upper bound theorem. Pages 3943 of: Convexity and related combinatorial geometry (Norman, Okla., 1980). Lecture Notes in Pure and Appl. Math., vol. 76. New York: Dekker. (393)Google Scholar
Brøndsted, A. 1983. An introduction to convex polytopes. Graduate Texts in Mathematics, vol. 90. New York: Springer-Verlag. (x, 32, 42, 70, 87, 88, 152, 153, 229, 393)Google Scholar
Brouwer, L. E. J. 1912. Beweis des Jordan’schen Satzes für den n-dimensionalen Raum. Math. Ann., 71, 314319. (401)CrossRefGoogle Scholar
Bruggesser, H., and Mani, P. 1971. Shellable decompositions of cells and spheres. Math. Scand., 29, 197205. (123, 128)CrossRefGoogle Scholar
Bui, H. T., Pineda-Villavicencio, G., and Ugon, J. 2024. The linkedness of cubical polytopes: Beyond the cube. Discrete Math., 347(3), 113801. (245)CrossRefGoogle Scholar
Bui, H. T., Pineda-Villavicencio, G., and Ugon, J. 2021. The linkedness of cubical polytopes: The cube. Electron. J. Comb., 28, P3.45. (249, 253)CrossRefGoogle Scholar
Bunt, L. N. H. 1934. Bijdrage tot de theorie der convexe puntverzamelingen. Thesis, University of Groningen, Amsterdam. (43)Google Scholar
Cai, M.C. 1993. The number of vertices of degree k in a minimally k-edge-connected graph. J. Comb. Theory, Ser. B, 58(2), 225239. (415)CrossRefGoogle Scholar
Carathéodory, C. 1907. Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann., 64, 95115. (20, 42)CrossRefGoogle Scholar
Chartrand, G., and Stewart, M. J. 1969. The connectivity of line-graphs. Math. Ann., 182(3), 170174. (414)CrossRefGoogle Scholar
Chartrand, G., Kaugars, A., and Lick, D. R. 1972. Critically n-connected graphs. Proc. Am. Math. Soc., 32(1), 6368. (413)Google Scholar
Conforti, M., Cornuéjols, G., and Zambelli, G. 2014. Integer programming. Vol. 271. Cham: Springer. (58)CrossRefGoogle Scholar
Connelly, R., and Guest, S. D. 2022. Frameworks, tensegrities, and symmetry. Cambridge: Cambridge University Press. (351, 371, 392)Google Scholar
Cordovil, R., and Duchet, P. 2000. Cyclic polytopes and oriented matroids. Eur. J. Comb., 21(1), 4964. (101)CrossRefGoogle Scholar
Courdurier, M. 2006. On stars and links of shellable polytopal complexes. J. Comb. Theory, Ser. A, 113(4), 692697. (154)Google Scholar
Cremona, L. 1890. Graphical statics. Two treatises on the graphical calculus and reciprocal figures in graphical statics. Oxford: Clarendon Press. (198)Google Scholar
Criado, F., and Santos, F. 2017. The maximum diameter of pure simplicial complexes and pseudo-manifolds. Discrete Comput. Geom., 58(3), 643649. (338)CrossRefGoogle Scholar
Danaraj, G., and Klee, V. 1974. Shellings of spheres and polytopes. Duke Math. J., 41, 443451. (173, 229)CrossRefGoogle Scholar
Dancis, J. 1984. Triangulated n-manifolds are determined by their [n/2]+ 1-skeletons. Topol. Appl., 18(1), 1726. (263)CrossRefGoogle Scholar
Dantzig, G. B. 1963. Linear programming and extensions. Princeton, NJ: Princeton University Press. (306)Google Scholar
Davies, J., and Pfender, F. 2021. Edge-maximal graphs on orientable and some nonorientable surfaces. J. Graph Theory, 98(3), 405425. (420)CrossRefGoogle Scholar
Davis, C. 1954. Theory of positive linear dependence. Am. J. Math., 76(4), 733746. (34)CrossRefGoogle Scholar
De Loera, J. A., Kim, E. D., Onn, S., and Santos, F. 2009. Graphs of transportation polytopes. J. Comb. Theory, Ser. A, 116(8), 13061325. (110)CrossRefGoogle Scholar
De Loera, J. A., Rambau, J., and Santos, F. 2010. Triangulations: Structures for algorithms and applications. Algorithms and Computation in Mathematics, vol. 25. Berlin: Springer-Verlag. (154)CrossRefGoogle Scholar
de Verdière, Y. C. 1991. Un principe variationnel pour les empilements de cercles. Invent. Math., 104(1), 655669. (212)CrossRefGoogle Scholar
Dehn, M. 1905. Die Eulersche Formel im Zusammenhang mit dem Inhalt in der nichteuklidischen Geometrie. Math. Ann., 61, 561586. (131)CrossRefGoogle Scholar
Dehn, M. 1916. Über die Starrheit konvexer Polyeder. Math. Ann., 77, 466473. (364, 366, 367)CrossRefGoogle Scholar
Del Pia, A., and Michini, C. 2016. On the diameter of lattice polytopes. Discrete Comput. Geom., 55(3), 681687. (333, 335)CrossRefGoogle Scholar
Deo, S. 2018. Algebraic topology. A primer. 2nd ed. Vol. 27. New Delhi: Hindustan Book Agency; Singapore: Springer. (401)CrossRefGoogle Scholar
Develin, M. 2004. LP-orientations of cubes and crosspolytopes. Adv. Geom., 4(4), 459468. (175, 226)CrossRefGoogle Scholar
Deza, A., and Pournin, L. 2018. Improved bounds on the diameter of lattice polytopes. Acta Math. Hung., 154(2), 457469. (335)CrossRefGoogle Scholar
Deza, A., and Pournin, L. 2019. Diameter, decomposability, and Minkowski sums of polytopes. Can. Math. Bull., 62(4), 741755. (299, 302)CrossRefGoogle Scholar
Diestel, R. 2017. Graph theory. 5th ed. Graduate Texts in Mathematics, vol. 173. Berlin: Springer-Verlag. (xi, 165, 181, 182, 215, 255, 405, 408, 411, 412, 414, 415, 416, 417, 419, 420, 423, 424, 425)CrossRefGoogle Scholar
Dirac, G. A. 1960. In abstrakten Graphen vorhandene vollständige 4-Graphen und ihre Unterteilungen. Math. Nachr., 22, 6185. (417)CrossRefGoogle Scholar
Doolittle, J. 2018. A minimal counterexample to strengthening of Perles’s conjecture. arXiv:809.00662. (271)Google Scholar
Doolittle, J., Nevo, E., Pineda-Villavicencio, G., Ugon, J., and Yost, D. 2018. On the reconstruction of polytopes. Discrete Comput. Geom., 61(2), 285302. (272, 397)Google Scholar
Edelsbrunner, H. 2012. Algorithms in combinatorial geometry. 1st ed. Berlin: Springer. (153)Google Scholar
Eggleston, H. G., Grünbaum, Branko, and Klee, Victor. 1964. Some semicontinuity theorems for convex polytopes and cell-complexes. Comment. Math. Helv., 39, 165188. (105, 153)CrossRefGoogle Scholar
Ehrenborg, R., and Hetyei, G. 1995. Generalizations of Baxter’s theorem and cubical homology. J. Comb. Theory, Ser. A, 69(2), 233287. (227)CrossRefGoogle Scholar
Eisenbrand, F., Hähnle, N., Razborov, A., and Rothvoß, T. 2010. Diameter of polyhedra: Limits of abstraction. Math. Oper. Res., 35(4), 786794. (335, 336, 338, 339)Google Scholar
Elmasry, A., Mehlhorn, K., and Schmidt, J. M. 2013. Every DFS tree of a 3-connected graph contains a contractible edge. J. Graph Theory, 72(1–2), 112121. (412, 425)Google Scholar
Eppstein, D. n.d. Twenty-one proofs of Euler’s formula: V − E + F = 2. Accessed 22 September 2021. www.ics.uci.edu/~eppstein/junkyard/euler/. (418)Google Scholar
Euler, L. 1736. Solutio problematis ad geometriam situs pertinentis. Comment. Acad. Sci. U. Petrop., 8 (Reprinted in Opera Omnia Series Prima, Vol. 7. pp. 1-10, 1766.), 128140. (404)Google Scholar
Euler, L. 1758a. Demonstratio nonnullarum insignium proprietatum quibas solida hedris planis inclusa sunt praedita. Novi Comm. Acad. Sci. Imp. Petropol., 4, 140160. (130, 418)Google Scholar
Euler, L. 1758b. Elementa doctrinae solidorum. Novi Comm. Acad. Sci. Imp. Petropol., 4, 109140. (128, 130, 418)Google Scholar
Ewald, G. 1996. Combinatorial convexity and algebraic geometry. Graduate Texts in Mathematics, vol. 168. New York: Springer-Verlag. (393)CrossRefGoogle Scholar
Ewald, G., and Shephard, G. C. 1974. Stellar subdivisions of boundary complexes of convex polytopes. Math. Ann., 210, 716. (150)CrossRefGoogle Scholar
Farkas, J. 1898. Die algebraischen Grundlagen der Anwendungen des Fourier’schen Principes in der Mechanik. Ungar. Ber., 15, 2540. (152)Google Scholar
Farkas, J. 1901. Theorie der einfachen Ungleichungen. J. Reine Angew. Math., 124, 127. (152)Google Scholar
Fáry, I. 1948. On straight line representation of planar graphs. Acta Sci. Math., 11, 229233. (158)Google Scholar
Felsner, S., and Rote, G. 2019. On primal-dual circle representations. Pages 8:1–8:18 of: Fineman, Jeremy, and Mitzenmacher, Michael (eds), Simplicity in Algorithms (Proc. 2nd Symposium, San Diego, 2019). OpenAccess Series in Informatics (OASIcs), vol. 69. Schloss Dagstuhl–Leibniz-Zentrum für Informatik. (230)Google Scholar
Firsching, M. 2020. The complete enumeration of 4-polytopes and 3-spheres with nine vertices. Isr. J. Math., 240, 417441. (228, 307, 395)CrossRefGoogle Scholar
Fleming, B., and Karu, K. 2010. Hard Lefschetz theorem for simple polytopes. J. Algebr. Comb., 32(2), 227239. (393)CrossRefGoogle Scholar
Flores, A. 1934. Über n-dimensionale Komplexe, die im R2n+1 absolut selbstverschlungen sind. Ergeb. Math. Kolloq., 6, 47. (155, 225)Google Scholar
Fortuna, E., Frigerio, R., and Pardini, R. 2016. Projective geometry: Solved problems and theory review. Vol. 104. Switzerland: Springer. (42)CrossRefGoogle Scholar
Fourier, J. B. J. 1827. Analyse des travaux de l’Académie Royale des Sciences pendant l’année 1824. Partie mathématique. (46)Google Scholar
Francese, C., and Richeson, D. 2007. The flaw in Euler’s proof of his polyhedral formula. Am. Math. Mon., 114(4), 286296. (128)CrossRefGoogle Scholar
Franklin, P. 1934. A six color problem. J. Math. Phys., 13, 363369. (420)CrossRefGoogle Scholar
Friedman, E. J. 2009. Finding a simple polytope from its graph in polynomial time. Discrete Comput. Geom., 41(2), 249256. (256, 264, 265, 269)CrossRefGoogle Scholar
Friedmann, O. 2011. A subexponential lower bound for Zadeh’s pivoting rule for solving linear programs and games. Pages 192206 of: Integer programming and combinatoral optimization (Proc. 15th international conference, IPCO 2011, New York, 2011). Berlin: Springer. (306)Google Scholar
Fukuda, K. 2004. From the zonotope construction to the Minkowski addition of convex polytopes. J. Symbolic Comput., 38(4), 12611272. (303, 304)CrossRefGoogle Scholar
Fukuda, K. 2022. Frequently asked questions in polyhedral computation. https://people.inf.ethz.ch/fukudak/. (52)Google Scholar
Fusy, Éric. 2006. Counting d-polytopes with d + 3 vertices. Electron. J. Comb., 13(1), R23. (144)CrossRefGoogle Scholar
Gale, D. 1954. Irreducible convex sets. Pages 217218 of: Proceedings of the International Congress of Mathematics, vol. II. (278)Google Scholar
Gale, D. 1963. Neighborly and cyclic polytopes. Pages 225232 of: Proc. Sympos. Pure Math, vol. 7. (95)CrossRefGoogle Scholar
Gallet, M., Grasegger, G., Legerský, J., and Schicho, J. 2021. Combinatorics of Bricard’s octahedra. C. R., Math., Acad. Sci. Paris, 359(1), 738. (367)Google Scholar
Gallier, J. 2011. Geometric methods and applications. 2nd ed. Texts in Applied Mathematics, vol. 38. New York: Springer. (8, 42, 153)CrossRefGoogle Scholar
Gallivan, S. 1985. Disjoint edge paths between given vertices of a convex polytope. J. Comb. Theory, Ser. A, 39(1), 112115. (231, 247, 254, 255, 396)CrossRefGoogle Scholar
Gawrilow, E., and Joswig, M. 2000. Polymake: A framework for analyzing convex polytopes. Pages 4373 of: Kalai, G., Ziegler, G.M. (eds) Polytopes— Combinatorics and computation. DMV Sem., vol. 29. Basel: Birkhäuser. (262)CrossRefGoogle Scholar
Geelen, J. 2012. On how to draw a graph. www.math.uwaterloo.ca/~jfgeelen/Publications/tutte.pdf. (230)Google Scholar
Gluck, H. 1975. Almost all simply connected closed surfaces are rigid. Pages 225239 of: Geometric topology (Proc. Conf. Park City 1974), Lect. Notes Math. vol. 438 Berlin: Springer. (361)CrossRefGoogle Scholar
Goodman, J. E., O’Rourke, J., and Tóth, C. D. (eds). 2017. Handbook of discrete and computational geometry. 3rd ed. Boca Raton, FL: Chapman & Hall/CRC. (110, 272, 353, 393)Google Scholar
Graham, R. L., Knuth, D. E., and Patashnik, O. 1994. Concrete mathematics: A foundation for computer science. 2nd ed. New York: Addison-Wesley. (375)Google Scholar
Graver, J. E. 2001. Counting on frameworks: Mathematics to aid the design of rigid structures. Vol. 25 of The Dolciani Mathematical Expositions, Washington, DC: Mathematical Association of America. (392)CrossRefGoogle Scholar
Graver, J., Servatius, B., and Servatius, H. 1993. Combinatorial rigidity. Providence, RI: American Mathematical Society. (392)CrossRefGoogle Scholar
Gruber, P. M. 2007. Convex and discrete geometry. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336. Berlin: Springer. (x, 128, 154)Google Scholar
Grünbaum, B. 1963. Unambiguous polyhedral graphs. Isr. J. Math., 1, 235238. (101, 102, 104, 153)CrossRefGoogle Scholar
Grünbaum, B. 1965. On the facial structure of convex polytopes. Bull. Am. Math. Soc., 71, 559560. (220, 230)CrossRefGoogle Scholar
Grünbaum, B. 1969. Imbeddings of simplicial complexes. Comment. Math. Helv., 44, 502513. (225)CrossRefGoogle Scholar
Grünbaum, B. 2003. Convex polytopes. 2nd ed. Graduate Texts in Mathematics, vol. 221. New York: Springer-Verlag. (x, 101, 102, 131, 133, 135, 136, 142, 151, 153, 154, 212, 218, 220, 228, 230, 256, 257, 258, 272, 303, 320, 340, 341, 344, 378, 395)CrossRefGoogle Scholar
Grünbaum, B., and Shephard, G. C. 1987. Some problems on polyhedra. J. Geom., 29, 182189. (214)CrossRefGoogle Scholar
Haase, C., and Ziegler, G. M. 2002. Examples and counterexamples for the Perles conjecture. Discrete Comput. Geom., 28(1), 2944. (271)CrossRefGoogle Scholar
Halin, R. 1966. Zu einem Problem von B. Grünbaum. Arch. Math., 17, 566568. (220, 228, 395)CrossRefGoogle Scholar
Halin, R. 1969. A theorem on n-connected graphs. J. Comb. Theory, 7, 150154. (413)CrossRefGoogle Scholar
Halmos, P. R. 1974. Finite-dimensional vector spaces. 2nd ed. New York: Springer-Verlag. (42)CrossRefGoogle Scholar
Helly, Ed. 1923. Über Mengen konvexer Körper mit gemeinschaftlichen Punkten. Jahresber. Dtsch. Math.-Ver., 32, 175176. (42)Google Scholar
Hirata, T., Kubota, K., and Saito, O. 1984. A sufficient condition for a graph to be weakly k-linked. J. Comb. Theory, Ser. B, 36(1), 8594. (255)CrossRefGoogle Scholar
Hodge, W. V. D., and Pedoe, D. 1994. Methods of algebraic geometry (Cambridge Mathematical Library). Cambridge: Cambridge University Press. (10)CrossRefGoogle Scholar
Hoffman, A. J., and Kruskal, J. G. 1956. Integral boundary points of convex polyhedra. Ann. Math. Stud., 38, 223246. (58)Google Scholar
Holmes, B. 2018. On the diameter of dual graphs of Stanley-Reisner rings and Hirsch type bounds on abstractions of polytopes. Electron. J. Comb., 25(1), P1.60. (339)CrossRefGoogle Scholar
Holt, F., and Klee, V. 1999. A proof of the strict monotone 4-step conjecture. Pages 201216 of: Advances in discrete and computational geometry (Prof. Conf. 1996 AMS-IMS-SIAM South Hadley, MA, 1996). Contemp. Math., vol. 223. Providence, RI: Amer. Math. Soc. (253)Google Scholar
Holt, F. B., and Klee, V. 1998. Many polytopes meeting the conjectured Hirsch bound. Discrete Comput. Geom., 20(1), 117. (332, 339)CrossRefGoogle Scholar
Hopcroft, J. E., and Kahn, P. J. 1992. A paradigm for robust geometric algorithms. Algorithmica, 7(4), 339380. (211, 230)CrossRefGoogle Scholar
Höppner, A., and Ziegler, G. M. 2000. A census of flag-vectors of 4-polytopes. Pages 105110 of: Kalai, G., Ziegler, G.M. (eds) Polytopes—Combinatorics and computation. DMV Sem., vol. 29. Basel: Birkhäuser. (343)CrossRefGoogle Scholar
Huck, A. 1991. A sufficient condition for graphs to be weakly k-linked. Graphs Comb., 7(4), 323351. (250, 251)CrossRefGoogle Scholar
Ishizeki, T., and Takeuchi, F. 1999. Geometric shellings of 3-polytopes. Pages 132135 of: Proc. 11th Canadian Conference on Computational Geometry. (175)Google Scholar
Izmestiev, I. 2009. Projective background of the infinitesimal rigidity of frameworks. Geom. Dedicata, 140, 183203. (363)CrossRefGoogle Scholar
Jänich, K. 1984. Topology. Undergraduate Texts in Mathematics. New York: Springer-Verlag. (402)CrossRefGoogle Scholar
Jockusch, W. 1993. The lower and upper bound problems for cubical polytopes. Discrete Comput. Geom., 9(2), 159163. (391, 399)CrossRefGoogle Scholar
Jordan, C. 1882. Cours d’analyse de l’École Polytechnique. 3 vol. Paris: Gauthier-Villars. (157, 229, 257, 401)Google Scholar
Joswig, M. 2000. Reconstructing a non-simple polytope from its graph. Pages 167176 of: Kalai, G., Ziegler, G.M. (eds) Polytopes—Combinatorics and computation. DMV Sem., vol. 29. Basel: Birkhäuser. (269, 271)CrossRefGoogle Scholar
Joswig, M. 2002. Projectivities in simplicial complexes and colorings of simple polytopes. Math. Z., 240(2), 243259. (230)CrossRefGoogle Scholar
Joswig, M., and Ziegler, G. M. 2000. Neighborly cubical polytopes. Discrete Comput. Geom., 24(2–3), 325344. (391, 399)CrossRefGoogle Scholar
Joswig, M., Kaibel, V., Pfetsch, M. E., and Ziegler, G. M. 2001. Vertex-facet incidences of unbounded polyhedra. Adv. Geom., 1(1), 2336. (310)CrossRefGoogle Scholar
Joswig, M., Kaibel, V., and Körner, F. 2002. On the k-systems of a simple polytope. Isr. J. Math., 129, 109117. (226, 269)CrossRefGoogle Scholar
Kaibel, V. 2003. Reconstructing a simple polytope from its graph. Pages 105118 of: Jünger, M., Reinelt, G., Rinaldi, G. (eds) Combinatorial optimization—Eureka, you shrink! Lecture Notes in Comput. Sci., vol. 2570. Berlin: Springer. (264)CrossRefGoogle Scholar
Kalai, G. 1987. Rigidity and the lower bound theorem. I. Invent. Math., 88(1), 125151. (340, 350, 373, 374, 385, 389, 392)CrossRefGoogle Scholar
Kalai, G. 1988a. A new basis of polytopes. J. Comb. Theory, Ser. B, 49(2), 191209. (390)CrossRefGoogle Scholar
Kalai, G. 1988b. A simple way to tell a simple polytope from its graph. J. Comb. Theory, Ser. A, 49(2), 381383. (171, 256, 272)CrossRefGoogle Scholar
Kalai, G. 1990. On low-dimensional faces that high-dimensional polytopes must have. Combinatorica, 10(3), 271280. (179, 181, 229, 389)CrossRefGoogle Scholar
Kalai, G. 1992. Upper bounds for the diameter and height of graphs of convex polyhedra. Discrete Comput. Geom., 8(4), 363372. (335, 336)CrossRefGoogle Scholar
Kalai, G. 1997. Linear programming, the simplex algorithm and simple polytopes. Math. Programming, 79(1–3), 217233. (306)CrossRefGoogle Scholar
Kalai, G., and Kleitman, D. J. 1992. A quasi-polynomial bound for the diameter of graphs of polyhedra. Bull. Am. Math. Soc., New Ser., 26(2), 315316. (320, 336)Google Scholar
Kalai, G. (coordinator). 2010. Polymath 3: Polynomial Hirsch conjecture. http://gilkalai.wordpress.com/2010/09/29/polymath-3-polynomial-hirsch-conjecture. (338)Google Scholar
Kallay, M. 1979. Decomposability of convex polytopes. Ph.D. thesis, The Hebrew University of Jerusalem, Jerusalem. (295)Google Scholar
Kallay, M. 1982. Indecomposable polytopes. Isr. J. Math., 41(3), 235243. (286, 291, 297, 300, 303, 304)CrossRefGoogle Scholar
Kallay, M. 1984. Decomposability of polytopes is a projective invariant. Pages 191196 of: Convexity and graph theory (Jerusalem, 1981). North-Holland Math. Stud., vol. 87. Amsterdam: North-Holland. (302, 304)Google Scholar
Karavelas, M. I., and Tzanaki, E. 2016a. A geometric approach for the upper bound theorem for Minkowski sums of convex polytopes. Discrete Comput. Geom., 56(4), 9661017. (277, 304)CrossRefGoogle Scholar
Karavelas, M. I., and Tzanaki, E. 2016b. The maximum number of faces of the Minkowski sum of two convex polytopes. Discrete Comput. Geom., 55, 748785. (277, 304)CrossRefGoogle Scholar
Karmarkar, N. 1984. A new polynomial-time algorithm for linear programming. Combinatorica, 4, 373395. (306)CrossRefGoogle Scholar
Kawarabayashi, K., Kostochka, A., and Yu, G. 2006. On sufficient degree conditions for a graph to be k-linked. Comb. Probab. Comput., 15(5), 685694. (255)CrossRefGoogle Scholar
Kelmans, A. K. 1980. Concept of a vertex in a matroid and 3-connected graphs. J. Graph Theory, 4, 1319. (182)CrossRefGoogle Scholar
Khachiyan, L. G. 1979. A polynomial algorithm in linear programming. Dokl. Akad. Nauk SSSR, 244, 10931096. (52, 306)Google Scholar
Kim, E. D., and Santos, F. 2010a. Companion to “An update on the Hirsch conjecture”. arXiv:0912.4235. (339)Google Scholar
Kim, E. D., and Santos, F. 2010b. An update on the Hirsch conjecture. Jahresber. Dtsch. Math.-Ver., 112(2), 7398. (339)CrossRefGoogle Scholar
Klee, V. 1964. A combinatorial analogue of Poincaré’s duality theorem. Can. J. Math., 16, 517531. (131)CrossRefGoogle Scholar
Klee, V. 1964a. Diameters of polyhedral graphs. Can. J. Math., 16, 602614. (318, 335)Google Scholar
Klee, V. 1964b. On the number of vertices of a convex polytope. Can. J. Math., 16, 701720. (106, 153, 392)CrossRefGoogle Scholar
Klee, V. 1964c. A property of d-polyhedral graphs. J. Math. Mech., 13, 10391042. (253)Google Scholar
Klee, V., and Minty, G. J. 1972. How good is the simplex algorithm? Pages 159175 of: Inequalities III (Proc. 3rd Symp., Los Angeles 1969). (306)Google Scholar
Klee, V., and Walkup, D. W. 1967. The d-step conjecture for polyhedra of dimension d < 6. Acta Math., 117, 5378. (305, 306, 307, 308, 310, 311, 312, 317, 320, 335, 338, 339)CrossRefGoogle Scholar
Kleinschmidt, P., and Onn, S. 1992. On the diameter of convex polytopes. Discrete Math., 102(1), 7577. (333)CrossRefGoogle Scholar
Koebe, P. 1936. Kontaktprobleme der konformen Abbildung. Ber. Sächs. Akad. Wiss. Leipzig, Math.-phys. Kl. 88, 141164. (212)Google Scholar
König, D. 1936. Theorie der endlichen und unendlichen Graphen. Kombinatorische Topologie der Streckenkomplexe. XI + 258 S. Leipzig, Akademische Verlagsge-sellschaft (Mathematik in Monographien, Bd. 16). (410)Google Scholar
Krein, M., and Milman, D. 1940. On extreme points of regular convex sets. Stud. Math., 9, 133138. (43)CrossRefGoogle Scholar
Kriesell, M. 2002. A survey on contractible edges in graphs of a prescribed vertex connectivity. Graphs Comb., 18(1), 130. (427)CrossRefGoogle Scholar
Kuratowski, K. 1930. Sur le probleme des courbes gauches en topologie. Fundam. Math., 15, 271283. (403, 423)CrossRefGoogle Scholar
Kusunoki, T., and Murai, S. 2019. The numbers of edges of 5-polytopes with a given number of vertices. Ann. Comb., 23(1), 89101. (343)CrossRefGoogle Scholar
Larman, D. G. 1970. Paths on polytopes. Proc. Lond. Math. Soc., 20, 161178. (327, 336)CrossRefGoogle Scholar
Larman, D. G., and Mani, P. 1970. On the existence of certain configurations within graphs and the 1-skeletons of polytopes. Proc. London Math. Soc., 20, 144160. (220, 226, 228, 229, 231, 245, 247, 249, 254, 255, 395)CrossRefGoogle Scholar
Lauritzen, N. 2013. Undergraduate convexity: From Fourier and Motzkin to Kuhn and Tucker. Hackensack, NJ: World Scientific Publishing. (38, 42, 152)CrossRefGoogle Scholar
Lebesgue, H. 1911. Sur l’invariance du nombre de dimensions d’un espace et sur le théorème de M. Jordan relatif aux variétés formées. C. R. Acad. Sci., Paris, 152, 841843. (401)Google Scholar
Lee, C. W. 1991. Regular triangulations of convex polytopes. Pages 443456 of: Gritzmann, P. and Sturmfels, B. (eds) Applied geometry and discrete mathematics: The Victor Klee Festschrift. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4. (111, 153)Google Scholar
Lee, C. W. 2013. Polytopes: Course notes. www.ms.uky.edu/~lee/ma714fa13/notes.pdf. (392)Google Scholar
Ling, J. M. 2007. New Non-Linear Inequalities for Flag-Vectors of 4-Polytopes. Discrete Comput. Geom., 37(3), 455469. (388, 390)CrossRefGoogle Scholar
Liu, G. Z. 1990. Proof of a conjecture on matroid base graphs. Sci. China Ser. A, 33, 13291337. (426)Google Scholar
Lockeberg, E. R. 1977. Refinements in boundary complexes of polytopes. Ph.D. thesis, University College London. (227, 228, 230, 253, 395, 396)Google Scholar
Lovász, L. 2007. Combinatorial problems and exercises. 2nd ed. Providence, RI: AMS Chelsea Publishing. (415)Google Scholar
Lovász, L. 2019. Graphs and geometry. Providence, RI: Colloquium Publications. (194, 211, 212, 214, 230, 361, 362, 371, 392)CrossRefGoogle Scholar
Lutz, F. H. 2004. Small examples of nonconstructible simplicial balls and spheres. SIAM J. Discrete Math., 18(1), 103109. (393)CrossRefGoogle Scholar
Lutz, F. H. 2008. Combinatorial 3-manifolds with 10 vertices. Beitr. Algebra Geom., 49(1), 97106. (393)Google Scholar
MacLane, S. 1937. A combinatorial condition for planar graphs. Fundam. Math., 28, 2232. (181, 182)CrossRefGoogle Scholar
Mader, W. 1971. Minimale n-fach kantenzusammenhängende Graphen. Math. Ann., 191, 2128. (415)CrossRefGoogle Scholar
Mader, W. 1974. Kantendisjunkte Wege in Graphen. Monatsh. Math., 78, 395404. (415)CrossRefGoogle Scholar
Mader, W. 1985. Paths in graphs, reducing the edge-connectivity only by two. Graphs Comb., 1(1), 8189. (426)CrossRefGoogle Scholar
Mader, W. 1986. Kritisch n-fach kantenzusammenhängende Graphen. (Critically n-edge-connected graphs). J. Comb. Theory, Ser. B, 40, 152158. (415)CrossRefGoogle Scholar
Mader, W. 1995. On vertices of degree n in minimally n-edge-connected graphs. Comb. Probab. Comput., 4(1), 8195. (415)CrossRefGoogle Scholar
Maharry, J. 1999. An excluded minor theorem for the octahedron. J. Graph Theory, 31(2), 95100. (221, 228, 230, 395)3.0.CO;2-N>CrossRefGoogle Scholar
Maharry, J. 2000. A characterization of graphs with no cube minor. J. Comb. Theory, Ser. B, 80(2), 179201. (221, 230)CrossRefGoogle Scholar
Maksimenko, A. M. 2009. The diameter of the ridge-graph of a cyclic polytope. Discrete Math. Appl., 19(1), 4753. (335)CrossRefGoogle Scholar
Marden, A., and Rodin, B. 1990. On Thurston’s formulation and proof of Andreev’s theorem. Pages 103115 of: Ruscheweyh, S., Saff, E. B., Salinas, L. C., and Varga, R. S. (eds), Computational methods and function theory (Proc. Conf., Valparaíso/Chile 1989). Lect. Notes Math., vol. 1435. Berlin: Springer. (212)CrossRefGoogle Scholar
Matoušek, J. 2002. Lectures on discrete geometry. Graduate Texts in Mathematics, vol. 212. New York: Springer-Verlag. (153)CrossRefGoogle Scholar
Matoušek, J. 2003. Using the Borsuk-Ulam theorem. Berlin: Springer-Verlag. (225)Google Scholar
Matschke, B., Santos, F., and Weibel, C. 2015. The width of five-dimensional prismatoids. Proc. Lond. Math. Soc., 110(3), 647672. (331, 332)CrossRefGoogle Scholar
Maxwell, J. C. 1864. XLV. On reciprocal figures and diagrams of forces. Philos. Mag, 27(182), 250261. (198)CrossRefGoogle Scholar
Maxwell, J. C. 1869. On reciprocal figures, frams, and diagrams of forces. Trans. R. Soc. Edinb., 26, 140. (198)CrossRefGoogle Scholar
McMullen, P. 1970. The maximum numbers of faces of a convex polytope. Mathematika, 17, 179184. (94, 153, 302, 340, 374, 375, 378, 393, 397)Google Scholar
McMullen, P. 1971a. The minimum number of facets of a convex polytope. J. Lond. Math. Soc., 3, 350354. (347)CrossRefGoogle Scholar
McMullen, P. 1971b. The numbers of faces of simplicial polytopes. Isr. J. Math., 9, 559570. (340, 377, 378)CrossRefGoogle Scholar
McMullen, P. 1976. Constructions for projectively unique polytopes. Discrete Math., 14(4), 347358. (150, 152)CrossRefGoogle Scholar
McMullen, P. 1987. Indecomposable convex polytopes. Isr. J. Math., 58(3), 321323. (289, 292, 304)Google Scholar
McMullen, P. 1993. On simple polytopes. Invent. Math., 113(2), 419444. (393)CrossRefGoogle Scholar
McMullen, P., and Shephard, G. C. 1971. Convex polytopes and the upper bound conjecture. London: Cambridge University Press. (141, 142, 153, 154)CrossRefGoogle Scholar
McMullen, P., and Walkup, D. W. 1971. A generalized lower-bound conjecture for simplicial polytopes. Mathematika, 18, 264273. (372, 374)CrossRefGoogle Scholar
Menger, K. 1927. Zur allgemeinen Kurventheorie. Fundam. Math., 10(1), 96115. (403, 415)CrossRefGoogle Scholar
Mészáros, G. 2015. Linkedness and path-pairability in the Cartesian product of graphs. Ph.D. thesis, Central European University. (253)Google Scholar
Mészáros, G. 2016. On linkedness in the Cartesian product of graphs. Period. Math. Hung., 72(2), 130138. (253)CrossRefGoogle Scholar
Meyer, W., and Kay, D. C. 1973. A convexity structure admits but one real linearization of dimension greater than one. J. Lond. Math. Soc., II. Ser., 7, 124130. (22)CrossRefGoogle Scholar
Meyer, W. J. 1969. Minkowski addition of convex sets. Ph.D. thesis, The University of Wisconsin. (297, 300, 301, 304)Google Scholar
Mihalisin, J., and Klee, V. 2000. Convex and linear orientations of polytopal graphs. Discrete Comput. Geom., 24(2–3), 421435. (174)CrossRefGoogle Scholar
Minkowski, H. 1896. Geometrie der zahlen. Leipzig: Teubner. (43, 152)Google Scholar
Minkowski, H. 1911. Gesammelte Abhandlungen von Hermann Minkowski. Unter Mitwirkung von Andreas Speiser und Hermann Weyl, herausgegeben von David Hilbert. Band I, II. Leipzig: Teubner. (43)Google Scholar
Mohar, B. 1993. A polynomial time circle packing algorithm. Discrete Math., 117(1), 257263. (214)CrossRefGoogle Scholar
Mohar, B., and Thomassen, C. 2001. Graphs on surfaces. Baltimore, MD: Johns Hopkins University Press. (157, 158, 182, 212, 214, 229, 230)CrossRefGoogle Scholar
Morris, S. A. 2020. Topology without tears. www.topologywithouttears.net/topbook.pdf. (402)Google Scholar
Motzkin, T. 1935. Sur quelques propriétés caractéristiques des ensembles convexes. Atti Accad. Naz. Lincei, Rend., VI. Ser., 21, 562567. (43)Google Scholar
Motzkin, T. 1936. Beiträge zur Theorie der linearen Ungleichungen. Thesis, University of Basel. (46, 152)Google Scholar
Motzkin, T. S. 1957. Comonotone curves and polyhedra. Bull. Am. Math. Soc., 63. (374)Google Scholar
Mulmuley, K. 1993. Computational geometry: An introduction through randomized algorithms. Englewood Cliffs, NJ: Prentice Hall. (393)Google Scholar
Murai, S., and Nevo, E. 2013. On the generalized lower bound conjecture for polytopes and spheres. Acta Math., 210(1), 185202. (374)CrossRefGoogle Scholar
Naddef, D. 1989. The Hirsch conjecture is true for (0,1)-polytopes. Math. Program., 45, 109110. (333)CrossRefGoogle Scholar
Nevo, E. 2022. Private Communication. (258)Google Scholar
Nevo, E., Pineda-Villavicencio, G., Ugon, J., and Yost, D. 2019. Almost simplicial polytopes I. The lower and upper bound theorems. Can. J. Math., 72(2). (393)Google Scholar
Okamura, H. 1984. Multicommodity flows in graphs. II. Jpn. J. Math., New Ser., 10(1), 99116. (255)CrossRefGoogle Scholar
Pach, J., and Agarwal, P. K. 1995. Combinatorial geometry. New York: John Wiley & Sons. (212)CrossRefGoogle Scholar
Pak, I. 2010. Lectures on discrete and polyhedral geometry. www.math.ucla.edu/~pak/book.htm. (392)Google Scholar
Papadakis, S. A., and Petrotou, V. 2020. The characteristic 2 anisotropicity of simplicial spheres. arXiv:2012.09815. (393)Google Scholar
Perles, M. A., and Prabhu, N. 1993. A property of graphs of convex polytopes. J. Comb. Theory, Ser. A, 62(1), 155157. (235, 252)CrossRefGoogle Scholar
Perles, M. A., and Shephard, G. C. 1967. Facets and nonfacets of convex polytopes. Acta Math., 119, 113145. (389, 391, 399)CrossRefGoogle Scholar
Perles, M. A., Martini, H., and Kupitz, Y. S. 2009. A Jordan–Brouwer separation theorem for polyhedral pseudomanifolds. Discrete Comput. Geom., 42(2), 277304. (402)CrossRefGoogle Scholar
Pestenjak, B. 1999. An algorithm for drawing planar graphs. Softw. Pract. Exper., 29(11), 973984. (207)Google Scholar
Pfeifle, J., Pilaud, V., and Santos, F. 2012. Polytopality and Cartesian products of graphs. Isr. J. Math., 192(1), 121141. (272)CrossRefGoogle Scholar
Pilaud, V., Pineda-Villavicencio, G., and Ugon, J. 2023. Edge connectivity of simplicial polytopes. Eur. J. Comb., 113, 103752. (231, 236, 237)CrossRefGoogle Scholar
Pineda-Villavicencio, G. 2021. A new proof of Balinski’s theorem on the connectivity of polytopes. Discrete Math., 344, 112408. (254, 427)CrossRefGoogle Scholar
Pineda-Villavicencio, G. 2022. Cycle space of graphs of polytopes. arXiv:2208.02579. (229)Google Scholar
Pineda-Villavicencio, G., and Schröter, B. 2022. Reconstructibility of matroid polytopes. SIAM J. Discrete Math., 36(1), 490508. (269)CrossRefGoogle Scholar
Pineda-Villavicencio, G., and Ugon, J. 2018. Polymake script to construct polytopes with isomorphic skeletons. http://guillermo.com.au/pdfs/ConstructPolytope-A.txt. (263)Google Scholar
Pineda-Villavicencio, G., and Yost, D. 2022. A lower bound theorem for d-polytopes with 2d +1 vertices. SIAM J. Discrete Math., 36, 29202941. (168, 350, 391, 399)CrossRefGoogle Scholar
Pineda-Villavicencio, G., Ugon, J., and Yost, D. 2018. The excess degree of a polytope. SIAM J. Discrete Math., 32(3), 20112046. (168, 169, 229, 296, 304, 343)CrossRefGoogle Scholar
Pineda-Villavicencio, G., Ugon, J., and Yost, D. 2019. Lower bound theorems for general polytopes. Eur. J. Comb., 79, 2745. (344)Google Scholar
Pineda-Villavicencio, G., Ugon, J., and Bui, H. T. 2020a. Connectivity of cubical polytopes. J. Combin. Theory Ser. A, 169, 105126. (254)Google Scholar
Pineda-Villavicencio, G., Ugon, J., and Yost, D. 2020b. Polytopes close to being simple. Discrete Comput. Geom., 64, 200215. (270)CrossRefGoogle Scholar
Pineda-Villavicencio, G., Ugon, J., and Yost, D. 2022. Minimum number of edges of d-polytopes with 2d +2 vertices. Electron. J. Comb., 29(3), P3.18. (168, 350, 391, 399)CrossRefGoogle Scholar
Pisanski, T., and Žitnik, A. 2009. Representing graphs and maps. Pages 151180 of: Gross, J., and Tucker, T. (authors) and Beineke, L., and Wilson, R. (eds), Topics in topological graph theory. Cambridge: Cambridge University Press. (207)CrossRefGoogle Scholar
Poincaré, H. 1893. Sur la généralisation d’un théorème d’Euler relatif aux polyèdres. C. R. Acad. Sci., Paris, 117, 144145. (128)Google Scholar
Preparata, F. P., and Shamos, M. I. 1985. Computational geometry: An introduction. Berlin, Heidelberg: Springer-Verlag. (153)CrossRefGoogle Scholar
Przesławski, K., and Yost, D. 2008. Decomposability of polytopes. Discrete Comput. Geom., 39(1–3), 460468. (303, 304)CrossRefGoogle Scholar
Przesławski, K., and Yost, D. 2016. More indecomposable polytopes. Extr. Math., 31, 169188. (290, 295, 298, 303, 304)Google Scholar
Radon, J. 1921. Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten. Math. Ann., 83, 113115. (20, 42)CrossRefGoogle Scholar
Reid, M., and Szendroi, B. 2005. Geometry and topology. Cambridge: Cambridge University Press. (42)CrossRefGoogle Scholar
Ribó Mor, A., Rote, G., and Schulz, A. 2011. Small grid embeddings of 3-polytopes. Discrete Comput. Geom., 45(1), 6587. (211)CrossRefGoogle Scholar
Richter-Gebert, J. 2006. Realization spaces of polytopes. Berlin: Springer. (64, 197, 211, 228, 230, 395)Google Scholar
Robertson, N., and Seymour, P. D. 1995. Graph minors. XIII. The disjoint paths problem. J. Comb. Theory, Ser. B, 63(1), 65110. (240)CrossRefGoogle Scholar
Roshchina, V., Sang, T., and Yost, D. 2018. Compact convex sets with prescribed facial dimensions. Pages 167175 of: de Gier, J., Praeger, C. E., and Tao, T. (eds), 2016 MATRIX Annals. Cham: Springer. (33)CrossRefGoogle Scholar
Roth, B. 1981. Rigid and flexible frameworks. Am. Math. Mon., 88(1), 621. (364, 392)CrossRefGoogle Scholar
Rowlands, R. 2022. Reconstructing d-manifold subcomplexes of cubes from their (d{2 + 1)-skeletons. Discrete Comput. Geom., 67, 492502. (263)CrossRefGoogle Scholar
Sachs, H. 1994. Coin graphs, polyhedra, and conformal mapping. Discrete Math., 134(1), 133138. (212, 230)CrossRefGoogle Scholar
Sallee, G. T. 1967. Incidence graphs of convex polytopes. J. Combinatorial Theory, 2, 466506. (161, 166, 229, 238, 251, 252)CrossRefGoogle Scholar
Santos, F. 2012. A counterexample to the Hirsch conjecture. Ann. Math., 176(1), 383412. (107, 108, 153, 305, 306, 328, 329, 331, 332, 339)CrossRefGoogle Scholar
Santos, F. 2013. Recent progress on the combinatorial diameter of polytopes and simplicial complexes. Top, 21(3), 426460. (335, 336, 337, 338, 339)CrossRefGoogle Scholar
Santos, F., Stephen, T., and Thomas, H. 2012. Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids. Discrete Comput. Geom., 47(3), 569576. (331)CrossRefGoogle Scholar
Sanyal, R., and Ziegler, G. M. 2010. Construction and analysis of projected deformed products. Discrete Comput. Geom., 43(2), 412435. (263)CrossRefGoogle Scholar
Sarkaria, K. S. 1991. Kuratowski complexes. Topology, 30(1), 6776. (225)CrossRefGoogle Scholar
Schild, G. 1993. Some minimal nonembeddable complexes. Topology Appl., 53(2), 177185. (225)CrossRefGoogle Scholar
Schläfli, L. (1901) 1850–52. Theorie der vielfachen Kontinuität. Vol. 38. Basel: Birkhäuser. (128)CrossRefGoogle Scholar
Schlegel, V. 1883. Theorie der homogen zusammengesetzten Raumgebilde (On the theory of homogeneous composed space forms). Nova Acta, 44, 343459. (153)Google Scholar
Schneider, R. 2014. Convex bodies: The Brunn-Minkowski theory. 2nd ed. Vol. 151. Cambridge: Cambridge University Press. (278, 303)Google Scholar
Schramm, O. 1992. How to cage an egg. Invent. Math., 107(3), 543560. (230)CrossRefGoogle Scholar
Schrijver, A. 1986. Theory of linear and integer programming. New York: John Wiley & Sons. (152)Google Scholar
Seymour, P. D. 1980. Disjoint paths in graphs. Discrete Math., 29(3), 293309. (240)CrossRefGoogle Scholar
Shemer, I. 1982. Neighborly polytopes. Isr. J. Math., 43(4), 291314. (263)Google Scholar
Shephard, G. C. 1963. Decomposable convex polyhedra. Mathematika, 10, 8995. (278, 293, 300, 304)CrossRefGoogle Scholar
Shifrin, T., and Adams, M. 2011. Linear algebra: A geometric approach. 2nd ed. New York: W. H. Freeman. (42)Google Scholar
Silverman, R. 1973. Decomposition of plane convex sets. I. Pac. J. Math., 47, 521530. (278, 289)CrossRefGoogle Scholar
Smilansky, Z. 1986a. Decomposability of polytopes and polyhedra. Ph.D. thesis, Hebrew University of Jerusalem. (297, 301)Google Scholar
Smilansky, Z. 1986b. An indecomposable polytope all of whose facets are decomposable. Mathematika, 33(2), 192196. (301, 303, 397)CrossRefGoogle Scholar
Smilansky, Z. 1987. Decomposability of polytopes and polyhedra. Geom. Dedicata, 24(1), 2949. (292, 300, 301, 302, 304, 397)CrossRefGoogle Scholar
Smilansky, Z. 1990. A nongeometric shelling of a 3-polytope. Isr. J. Math., 71(1), 2932. (175)CrossRefGoogle Scholar
Soltan, V. 2015. Lectures on convex sets. Hackensack: World Scientific. (42)CrossRefGoogle Scholar
Sommerville, D. M. Y. 1927. The relations connecting the angle-sums and volume of a polytope in space of n dimensions. Proc. R. Soc. Lond., A, 115(770), 103119. (131)Google Scholar
Sommerville, D. M. Y. 1958. An introduction to the geometry of n dimensions. New York: Dover Publications. (153)Google Scholar
Spielman, D. A. 2019. Spectral and algebraic graph theory. http://cs-www.cs.yale.edu/homes/spielman/sagt/sagt.pdf. (194, 230)Google Scholar
Stanley, R. P. 1975. The upper bound conjecture and Cohen-Macaulay rings. Stud. Appl. Math., 54, 135142. (393)CrossRefGoogle Scholar
Stanley, R. P. 1980. The number of faces of a simplicial convex polytope. Adv. Math., 35, 236238. (378, 393)CrossRefGoogle Scholar
Stanley, R. P. 1996. Combinatorics and commutative algebra. 2nd ed. Progress in Mathematics, vol. 41. Boston, MA: Birkhäuser. (393)Google Scholar
Steinitz, E. 1906. Über die Eulerschen Polyederrelationen. Arch. der Math. u. Phys., 11, 8688. (340, 341)Google Scholar
Steinitz, E. 1922. Polyeder und raumeinteilungen. Encyclop. d. math. Wiss., 3, 1139. (198, 211)Google Scholar
Sturmfels, B. 1987. Cyclic polytopes and d-order curves. Geom. Dedicata, 24(1), 103107. (100)CrossRefGoogle Scholar
Sturmfels, B. 1988. Some applications of affine Gale diagrams to polytopes with few vertices. SIAM J. Discrete Math., 1(1), 121133. (145, 212)CrossRefGoogle Scholar
Sturmfels, B. 1996. Gröbner bases and convex polytopes. Vol. 8. Providence, RI: American Mathematical Society. (110)Google Scholar
Sukegawa, N. 2019. An asymptotically improved upper bound on the diameter of polyhedra. Discrete Comput. Geom., 62(3), 690699. (323)CrossRefGoogle Scholar
Tay, T.-S. 1995. Lower-bound theorems for pseudomanifolds. Discrete Comput. Geom., 13(2), 203216. (374)CrossRefGoogle Scholar
Thomas, R., and Wollan, P. 2005. An improved linear edge bound for graph linkages. Eur. J. Comb., 26(3–4), 309324. (249, 255)CrossRefGoogle Scholar
Thomas, R., and Wollan, P. 2008. The extremal function for 3-linked graphs. J. Comb. Theory, Ser. B, 98(5), 939971. (254, 396)CrossRefGoogle Scholar
Thomassen, C. 1980a. 2-linked graphs. Eur. J. Comb., 1(4), 371378. (240, 249, 250, 254, 396, 412, 427)CrossRefGoogle Scholar
Thomassen, C. 1980b. Planarity and duality of finite and infinite graphs. J. Comb. Theory, Ser. B, 29(2), 244271. (423)CrossRefGoogle Scholar
Thomassen, C. 1981. Nonseparating cycles in k-connected graphs. J. Graph Theory, 5, 351354. (426)CrossRefGoogle Scholar
Thurston, W. P. 1980. The geometry and topology of three-manifolds. Lecture notes at Princeton University, http://library.msri.org/books/gt3m/PDF/Thurston-gt3m.pdf. (212)Google Scholar
Timan, A. F. 1963. Theory of approximation of functions of a real variable. New York: Pergamon Press. (100)Google Scholar
Todd, M. J. 1980. The monotonic bounded Hirsch conjecture is false for dimension at least 4. Math. Oper. Res., 5, 599601. (307, 309, 310)CrossRefGoogle Scholar
Todd, M. J. 2002. The many facets of linear programming. Math. Program., 91(3), 417436. (306)CrossRefGoogle Scholar
Todd, M. J. 2014. An improved Kalai-Kleitman bound for the diameter of a polyhedron,. SIAM J. Discrete Math., 28, 19441947. (319, 320, 322, 339)CrossRefGoogle Scholar
Tutte, W. T. 1963. How to draw a graph. Proc. Lond. Math. Soc., 13, 743768. (155, 164, 165, 182, 184, 186, 214, 230)CrossRefGoogle Scholar
Tverberg, H. 1966. A generalization of Radon’s theorem. J. Lond. Math. Soc., 41, 123128. (42)CrossRefGoogle Scholar
Van Kampen, E. R. 1932. Komplexe in euklidischen Räumen. Abh. Math. Semin. Univ. Hamb., 9, 7278. (155, 225)CrossRefGoogle Scholar
Wagner, K. 1937. Über eine Eigenschaft der ebenen Komplexe. Math. Ann., 114, 570590. (423)CrossRefGoogle Scholar
Walkup, D. W. 1970. The lower bound conjecture for 3- and 4-manifolds. Acta Math., 125, 75107. (372)CrossRefGoogle Scholar
Webster, R. 1994. Convexity. New York: Oxford University Press. (xi, 3, 20, 21, 23, 24, 25, 27, 28, 32, 35, 37, 38, 42, 130, 140, 141, 144, 152, 153, 154, 229, 402)CrossRefGoogle Scholar
Werner, A., and Wotzlaw, R. F. 2011. On linkages in polytope graphs. Adv. Geom., 11(3), 411427. (231, 245, 249, 254, 255, 396)CrossRefGoogle Scholar
West, D. B. 2001. Introduction to graph theory. 2nd ed. Upper Saddle River, NJ: Prentice Hall. (414, 427)Google Scholar
Weyl, H. 1935. Elementare theorie der konvexen polyeder. Comment. Math. Helv., 7, 290306. (152)CrossRefGoogle Scholar
Whiteley, W. 1984. Infinitesimally rigid polyhedra. I. Statics of frameworks. Trans. Amer. Math. Soc., 285(2), 431465. (367, 370)CrossRefGoogle Scholar
Whiteley, W. 1992. Matroids and rigid structures. Pages 153 of: White, N. (ed), Matroid Applications (Encyclopedia of Mathematics and its Applications). Cambridge: Cambridge University Press. (392)Google Scholar
Whiteley, W. 1996. Some matroids from discrete applied geometry. Pages 171311 of: Matroid theory (Seattle, WA, 1995). Contemp. Math., vol. 197. Providence, RI: Amer. Math. Soc. (388)Google Scholar
Whitney, H. 1932. Congruent graphs and the connectivity of graphs. Am. J. Math., 54(1), 150168. (414, 417, 426)CrossRefGoogle Scholar
Whitney, H. 1932. Non-separable and planar graphs. Trans. Am. Math. Soc., 34, 339362. (403, 423, 424)CrossRefGoogle Scholar
Whitney, H. 1933. 2-isomorphic graphs. Am. J. Math., 55, 245254. (164)CrossRefGoogle Scholar
Williamson Hoke, K. 1988. Completely unimodal numberings of a simple polytope. Discrete Appl. Math., 20(1), 6981. (226)CrossRefGoogle Scholar
Wotzlaw, R. F. 2009. Incidence graphs and unneighborly polytopes. Ph.D. thesis, Technical University of Berlin. (230, 252)Google Scholar
Xue, L. 2021. A proof of Grünbaum’s lower bound conjecture for general polytopes. Isr. J. Math., 245, 9911000. (340, 344, 392)CrossRefGoogle Scholar
Xue, L. 2022. A lower bound theorem for strongly regular CW spheres up to 2d + 1. arXiv:2207.13839. (350, 388)CrossRefGoogle Scholar
Yaglom, I. M. 1973. Geometric transformations III: Affine and projective transformations. New York: Random House. (153)CrossRefGoogle Scholar
Yang, Y. 2021. A note on the diameter of convex polytope. Discrete Appl. Math., 289, 534538. (335)CrossRefGoogle Scholar
Yost, D. 2007. Some indecomposable polyhedra. Optimization, 56(5–6), 715724. (303)CrossRefGoogle Scholar
Ziegler, G. M. 1995. Lectures on polytopes. Graduate Texts in Mathematics, vol. 152. New York: Springer-Verlag. (x, 46, 50, 74, 75, 102, 114, 123, 134, 136, 149, 152, 153, 154, 174, 212, 229, 303, 331, 338, 393, 394)CrossRefGoogle Scholar
Ziegler, G. M. 2000. Lectures on 0{1-polytopes. Pages 141 of: Kalai, G., Ziegler, G.M. (eds) Polytopes—Combinatorics and computation. DMV Sem., vol. 29. Basel: Birkhäuser. (333)Google Scholar
Ziegler, G. M. 2007. Convex polytopes: extremal constructions and f -vector shapes. Pages 617691 of: Miller, E., Reiner, V., and Sturmfels, B. (eds), Geometric combinatorics. IAS/Park City Math. Ser., vol. 13. Providence, RI: Amer. Math. Soc. (211)CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×