{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:46:12Z","timestamp":1740109572758,"version":"3.37.3"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2019,11,4]],"date-time":"2019-11-04T00:00:00Z","timestamp":1572825600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,11,4]],"date-time":"2019-11-04T00:00:00Z","timestamp":1572825600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"Austrian Science Fund","award":["J-3847-N35"],"award-info":[{"award-number":["J-3847-N35"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2020,10]]},"DOI":"10.1007\/s00454-019-00147-1","type":"journal-article","created":{"date-parts":[[2019,11,4]],"date-time":"2019-11-04T18:03:08Z","timestamp":1572890588000},"page":"1067-1097","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices"],"prefix":"10.1007","volume":"64","author":[{"given":"Alexander","family":"Pilz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Emo","family":"Welzl","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5360-265X","authenticated-orcid":false,"given":"Manuel","family":"Wettstein","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,11,4]]},"reference":[{"issue":"3","key":"147_CR1","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/s00373-005-0612-5","volume":"21","author":"BM \u00c1brego","year":"2005","unstructured":"\u00c1brego, B.M., Fern\u00e1ndez-Merchant, S.: A lower bound for the rectilinear crossing number. Graphs Comb. 21(3), 293\u2013300 (2005)","journal-title":"Graphs Comb."},{"key":"147_CR2","unstructured":"Afshani, P., Sheehy, D.R., Stein, Y.: Approximating the simplicial depth (2015). \narXiv:1512.04856"},{"issue":"Supplement\u20131","key":"147_CR3","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s00373-007-0704-5","volume":"23","author":"O Aichholzer","year":"2007","unstructured":"Aichholzer, O., Hackl, T., Huemer, C., Hurtado, F., Krasser, H., Vogtenhuber, B.: On the number of plane geometric graphs. Graphs Combin. 23(Supplement\u20131), 67\u201384 (2007)","journal-title":"Graphs Combin."},{"issue":"6","key":"147_CR4","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1006\/eujc.1998.0235","volume":"19","author":"M Aigner","year":"1998","unstructured":"Aigner, M.: Motzkin numbers. Eur. J. Combin. 19(6), 663\u2013675 (1998)","journal-title":"Eur. J. Combin."},{"issue":"5","key":"147_CR5","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1007\/BF01769706","volume":"10","author":"EM Arkin","year":"1993","unstructured":"Arkin, E.M., Khuller, S., Mitchell, J.S.B.: Geometric knapsack problems. Algorithmica 10(5), 399\u2013427 (1993)","journal-title":"Algorithmica"},{"key":"147_CR6","first-page":"39","volume":"58","author":"HW Becker","year":"1952","unstructured":"Becker, H.W.: Planar rhyme schemes. Bull. Am. Math. Soc. 58, 39 (1952)","journal-title":"Bull. Am. Math. Soc."},{"key":"147_CR7","unstructured":"Brouwer, A.E.: The enumeration of locally transitive tournaments. Technical Report Report ZW 138\/80, Mathematisch Centrum, Amsterdam (1980)"},{"key":"147_CR8","volume-title":"Algebraic Complexity Theory. Grundlehren der Mathematischen Wissenschaften.","author":"P B\u00fcrgisser","year":"1996","unstructured":"B\u00fcrgisser, P., Lickteig, T., Clausen, M., Shokrollahi, A.: Algebraic Complexity Theory. Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1996)"},{"key":"147_CR9","unstructured":"Cheng, A.Y., Ouyang, M.: On algorithms for simplicial depth. In: Proceedings of the 13th Canadian Conference on Computational Geometry, pp. 53\u201356 (2001)"},{"issue":"1","key":"147_CR10","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/0012-365X(93)90326-O","volume":"117","author":"S Dulucq","year":"1993","unstructured":"Dulucq, S., Penaud, J.-G.: Cordes, arbres et permutations. Discret. Math. 117(1), 89\u2013105 (1993)","journal-title":"Discret. Math."},{"key":"147_CR11","unstructured":"Dumitrescu, A., Schulz, A., Sheffer, A., T\u00f3th, Cs.D.: Bounds on the maximum multiplicity of some common geometric graphs. SIAM J. Discret. Math. 27(2), 802\u2013826 (2013)"},{"issue":"1","key":"147_CR12","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1145\/77635.77639","volume":"9","author":"H Edelsbrunner","year":"1990","unstructured":"Edelsbrunner, H., M\u00fccke, E.P.: Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graph. 9(1), 66\u2013104 (1990)","journal-title":"ACM Trans. Graph."},{"issue":"2","key":"147_CR13","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1137\/0215024","volume":"15","author":"H Edelsbrunner","year":"1986","unstructured":"Edelsbrunner, H., O\u2019Rourke, J., Seidel, R.: Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput. 15(2), 341\u2013363 (1986)","journal-title":"SIAM J. Comput."},{"issue":"1","key":"147_CR14","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/BF02187823","volume":"7","author":"D Eppstein","year":"1992","unstructured":"Eppstein, D., Overmars, M.H., Rote, G., Woeginger, G.J.: Finding minimum area $$k$$-gons. Discrete Comput. Geom. 7(1), 45\u201358 (1992)","journal-title":"Discrete Comput. Geom."},{"issue":"1\u20133","key":"147_CR15","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/S0012-365X(98)00372-0","volume":"204","author":"P Flajolet","year":"1999","unstructured":"Flajolet, P., Noy, M.: Analytic combinatorics of non-crossing configurations. Discrete Math. 204(1\u20133), 203\u2013229 (1999)","journal-title":"Discrete Math."},{"issue":"1\u20133","key":"147_CR16","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/0012-365X(92)90658-3","volume":"108","author":"J Gil","year":"1992","unstructured":"Gil, J., Steiger, W.L., Wigderson, A.: Geometric medians. Discrete Math. 108(1\u20133), 37\u201351 (1992)","journal-title":"Discrete Math."},{"issue":"3","key":"147_CR17","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1137\/0212032","volume":"12","author":"JE Goodman","year":"1983","unstructured":"Goodman, J.E., Pollack, R.: Multidimensional sorting. SIAM J. Comput. 12(3), 484\u2013507 (1983)","journal-title":"SIAM J. Comput."},{"issue":"3","key":"147_CR18","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/0097-3165(84)90050-5","volume":"37","author":"JE Goodman","year":"1984","unstructured":"Goodman, J.E., Pollack, R.: Semispaces of configurations, cell complexes of arrangements. J. Combin. Theory Ser. A 37(3), 257\u2013293 (1984)","journal-title":"J. Combin. Theory Ser. A"},{"key":"147_CR19","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4613-0019-9","volume-title":"Convex Polytopes","author":"B Gr\u00fcnbaum","year":"2003","unstructured":"Gr\u00fcnbaum, B.: Convex Polytopes, 2nd edn. Springer, Berlin (2003)","edition":"2"},{"key":"147_CR20","unstructured":"Huemer, C., Pilz, A., Silveira, R.I.: A new lower bound on the maximum number of plane graphs using production matrices. In: Proceedings of the 34th European Workshop on Computational Geometry, pp. 9:1\u20139:6 (2018)"},{"key":"147_CR21","first-page":"169","volume":"45","author":"F Hurtado","year":"1997","unstructured":"Hurtado, F., Noy, M.: Counting triangulations of almost-convex polygons. Ars Combin. 45, 169\u2013179 (1997)","journal-title":"Ars Combin."},{"issue":"6","key":"147_CR22","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/0020-0190(90)90217-L","volume":"33","author":"S Khuller","year":"1990","unstructured":"Khuller, S., Mitchell, J.S.B.: On a triangle counting problem. Inf. Process. Lett. 33(6), 319\u2013321 (1990)","journal-title":"Inf. Process. Lett."},{"issue":"2","key":"147_CR23","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/s004930050055","volume":"19","author":"S Linusson","year":"1999","unstructured":"Linusson, S.: The number of $$M$$-sequences and $$f$$-vectors. Combinatorica 19(2), 255\u2013266 (1999)","journal-title":"Combinatorica"},{"issue":"1","key":"147_CR24","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1214\/aos\/1176347507","volume":"18","author":"RY Liu","year":"1990","unstructured":"Liu, R.Y.: On a notion of data depth based on random simplices. Ann. Stat. 18(1), 405\u2013414 (1990)","journal-title":"Ann. Stat."},{"key":"147_CR25","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1090\/conm\/342\/06138","volume-title":"Towards a Theory of Geometric Graphs. Contemporary Mathematics","author":"L Lov\u00e1sz","year":"2004","unstructured":"Lov\u00e1sz, L., Vesztergombi, K., Wagner, U., Welzl, E.: Convex quadrilaterals and $$k$$-sets. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 342, pp. 139\u2013148. American Mathematical Society, Providence (2004)"},{"key":"147_CR26","doi-asserted-by":"crossref","unstructured":"Matou\u0161ek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)","DOI":"10.1007\/978-1-4613-0039-7"},{"issue":"1","key":"147_CR27","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.jcta.2004.01.005","volume":"106","author":"JJ Montellano-Ballesteros","year":"2004","unstructured":"Montellano-Ballesteros, J.J., Strausz, R.: Counting polytopes via the Radon complex. J. Comb. Theory Ser. A 106(1), 109\u2013121 (2004)","journal-title":"J. Comb. Theory Ser. A"},{"key":"147_CR28","doi-asserted-by":"crossref","unstructured":"Motzkin, Th.: Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products. Bull. Am. Math. Soc. 54(4), 352\u2013360 (1948)","DOI":"10.1090\/S0002-9904-1948-09002-4"},{"issue":"1","key":"147_CR29","doi-asserted-by":"crossref","first-page":"203","DOI":"10.2140\/pjm.1984.110.203","volume":"110","author":"EM Palmer","year":"1984","unstructured":"Palmer, E.M., Robinson, R.W.: Enumeration of self-dual configurations. Pac. J. Math. 110(1), 203\u2013221 (1984)","journal-title":"Pac. J. Math."},{"key":"147_CR30","unstructured":"Randall, D., Rote, G., Santos, F., Snoeyink, J.: Counting triangulations and pseudo-triangulations of wheels. In: Proceedings of the 13th Canadian Conference on Computational Geometry, pp. 149\u2013152 (2001)"},{"issue":"4","key":"147_CR31","first-page":"516","volume":"45","author":"PJ Rousseeuw","year":"1996","unstructured":"Rousseeuw, P.J., Ruts, I.: Bivariate location depth. J. R. Stat. Soc. Ser. C 45(4), 516\u2013526 (1996)","journal-title":"J. R. Stat. Soc. Ser. C"},{"key":"147_CR32","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1007\/978-3-319-44479-6_30","volume-title":"A Journey Through Discrete Mathematics: A Tribute to Ji\u0159\u00ed Matou\u0161ek","author":"AJ Ruiz-Vargas","year":"2017","unstructured":"Ruiz-Vargas, A.J., Welzl, E.: Crossing-free perfect matchings in wheel point sets. In: Loebl, M., Ne\u0161et\u0159il, J., Thomas, R. (eds.) A Journey Through Discrete Mathematics: A Tribute to Ji\u0159\u00ed Matou\u0161ek, pp. 735\u2013764. Springer, Cham (2017)"},{"issue":"1","key":"147_CR33","doi-asserted-by":"crossref","first-page":"Art. No. 70","DOI":"10.37236\/557","volume":"18","author":"M Sharir","year":"2011","unstructured":"Sharir, M., Sheffer, A.: Counting triangulations of planar point sets. Electron. J. Comb. 18(1), Art. No. 70 (2011)","journal-title":"Electron. J. Comb."},{"issue":"3","key":"147_CR34","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1137\/050636036","volume":"36","author":"M Sharir","year":"2006","unstructured":"Sharir, M., Welzl, E.: On the number of crossing-free matchings, cycles, and partitions. SIAM J. Comput. 36(3), 695\u2013720 (2006)","journal-title":"SIAM J. Comput."},{"issue":"4","key":"147_CR35","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1016\/j.jcta.2013.01.002","volume":"120","author":"M Sharir","year":"2013","unstructured":"Sharir, M., Sheffer, A., Welzl, E.: Counting plane graphs: Perfect matchings, spanning cycles, and Kasteleyn\u2019s technique. J. Comb. Theory Ser. A 120(4), 777\u2013794 (2013)","journal-title":"J. Comb. Theory Ser. A"},{"key":"147_CR36","volume-title":"Oriented Projective Geometry","author":"J Stolfi","year":"1991","unstructured":"Stolfi, J.: Oriented Projective Geometry. Academic Press, Boston (1991)"},{"key":"147_CR37","unstructured":"Wagner, U.: On the rectilinear crossing number of complete graphs. In: Proceedings of the 14th Annual Symposium on Discrete Algorithms, pp. 583\u2013588. ACM\/SIAM, San Diego (2003)"},{"issue":"2","key":"147_CR38","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1007\/s00454-001-0028-9","volume":"26","author":"U Wagner","year":"2001","unstructured":"Wagner, U., Welzl, E.: A continuous analogue of the upper bound theorem. Discrete Comput. Geom. 26(2), 205\u2013219 (2001)","journal-title":"Discrete Comput. Geom."},{"issue":"3","key":"147_CR39","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1007\/s004540010085","volume":"25","author":"E Welzl","year":"2001","unstructured":"Welzl, E.: Entering and leaving $${j}$$-facets. Discrete Comput. Geom. 25(3), 351\u2013364 (2001)","journal-title":"Discrete Comput. Geom."},{"key":"147_CR40","doi-asserted-by":"crossref","unstructured":"Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)","DOI":"10.1007\/978-1-4613-8431-1"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-019-00147-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00454-019-00147-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-019-00147-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,11,3]],"date-time":"2020-11-03T00:11:18Z","timestamp":1604362278000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00454-019-00147-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,4]]},"references-count":40,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,10]]}},"alternative-id":["147"],"URL":"https:\/\/doi.org\/10.1007\/s00454-019-00147-1","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"type":"print","value":"0179-5376"},{"type":"electronic","value":"1432-0444"}],"subject":[],"published":{"date-parts":[[2019,11,4]]},"assertion":[{"value":"7 January 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 August 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 October 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 November 2019","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}