{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:39:09Z","timestamp":1787323149965,"version":"build-2736575974"},"reference-count":20,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/501100003141","name":"Consejo Nacional de Ciencia y Tecnolog\u00eda","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100003141","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010665","name":"H2020 Marie Sk\u0142odowska-Curie Actions","doi-asserted-by":"publisher","award":["754411"],"award-info":[{"award-number":["754411"]}],"id":[{"id":"10.13039\/100010665","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["50503-10940-500"],"award-info":[{"award-number":["50503-10940-500"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>Motivated by the successful application of geometry to proving the Harary--Hill conjecture for \u201cpseudolinear\u201d drawings of $K_n$, we introduce \u201cpseudospherical\u201d drawings of graphs. A spherical drawing of a graph $G$ is a drawing in the unit sphere $\\mathbb{S}^2$ in which the vertices of $G$ are represented as points---no three on a great circle---and the edges of $G$ are shortest-arcs in $\\mathbb{S}^2$ connecting pairs of vertices. Such a drawing has three properties: (1) every edge $e$ is contained in a simple closed curve $\\gamma_e$ such that the only vertices in $\\gamma_e$ are the ends of $e$; (2) if $e\\ne f$, then $\\gamma_e\\cap\\gamma_f$ has precisely two crossings; and (3) if $e\\ne f$, then $e$ intersects $\\gamma_f$ at most once, in either a crossing or an end of $e$. We use properties (1)--(3) to define a pseudospherical drawing of $G$. Our main result is that for the complete graph, properties (1)--(3) are equivalent to the same three properties but with \u201cprecisely two crossings\u201d in (2) replaced by \u201cat most two crossings.\u201d The proof requires a result in the geometric transversal theory of arrangements of pseudocircles. This is proved using the surprising result that the absence of special arcs (coherent spirals) in an arrangement of simple closed curves characterizes the fact that any two curves in the arrangement have at most two crossings. Our studies provide the necessary ideas for exhibiting a drawing of $K_{10}$ that has no extension to an arrangement of pseudocircles and a drawing of $K_9$ that does extend to an arrangement of pseudocircles, but no such extension has all pairs of pseudocircles crossing twice.<\/jats:p>","DOI":"10.1137\/20m1313234","type":"journal-article","created":{"date-parts":[[2021,5,20]],"date-time":"2021-05-20T11:53:17Z","timestamp":1621511597000},"page":"1050-1076","source":"Crossref","is-referenced-by-count":4,"title":["Extending Drawings of Complete Graphs into Arrangements of Pseudocircles"],"prefix":"10.1137","volume":"35","author":[{"given":"Alan","family":"Arroyo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R. Bruce","family":"Richter","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Matthew","family":"Sunohara","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,5,20]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/s00373-005-0612-5"},{"key":"atypb2","first-page":"57","author":"\u00c1brego B. M.","year":"2015","journal-title":"Slovenia"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/17M1147974"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-014-9635-0"},{"key":"atypb5","first-page":"46","author":"\u00c1brego B. M.","year":"2014","journal-title":"Nova Scotia"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-013-9514-0"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-007-1325-8"},{"key":"atypb8","first-page":"33","author":"Aichholzer O.","year":"2015","journal-title":"Barcelona"},{"key":"atypb9","first-page":"3","volume":"12","author":"Arroyo A.","year":"2021","journal-title":"J. Comput. Geom."},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.22167"},{"key":"atypb11","first-page":"131","volume":"67","author":"Arroyo A.","year":"2017","journal-title":"Australas. J. Combin."},{"key":"atypb13","unstructured":"A. Bj\u00f6rner, M. Las Vergnas, B. Sturmfelds, N. White, and G. Ziegler,\n                      Oriented Matroids\n                      , Encyclopedia Math. Appl. 46, Cambridge University Press, 2009."},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-019-00077-y"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1017\/S0013091500025645"},{"key":"atypb17","unstructured":"J. Kyn\u010dl\n                      Drawing of Complete Graphs with $Z(n)$ Crossing,\n                      https:\/\/mathoverflow.net\/questions\/128878\/drawings-of-complete-graphs-with-zn-crossings, 2013."},{"key":"atypb18","first-page":"139","author":"Lov\u00e1sz L.","year":"2004","journal-title":"Amer. Math. Soc."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1090\/proc\/14450"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/0113032"},{"key":"atypb21","unstructured":"N. H. Rafla,\n                      The Good Drawings $D_n$ of the Complete Graph $K_n$\n                      , Ph.D. thesis, McGill University, Montreal, 1988."},{"key":"atypb22","first-page":"309","author":"Snoeyink J.","year":"1991","journal-title":"Amer. Math. Soc."}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1313234","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:44:20Z","timestamp":1787319860000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1313234"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":20,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/20M1313234"],"URL":"https:\/\/doi.org\/10.1137\/20m1313234","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}