{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T06:39:12Z","timestamp":1774334352533,"version":"3.50.1"},"publisher-location":"Singapore","reference-count":35,"publisher":"Springer Nature Singapore","isbn-type":[{"value":"9789819571260","type":"print"},{"value":"9789819571277","type":"electronic"}],"license":[{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026]]},"DOI":"10.1007\/978-981-95-7127-7_1","type":"book-chapter","created":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T10:06:55Z","timestamp":1770977215000},"page":"3-18","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Computing Beyond-Planar Crossing Numbers via\u00a0Forbidden Crossing Patterns"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4681-5550","authenticated-orcid":false,"given":"Markus","family":"Chimani","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4593-8740","authenticated-orcid":false,"given":"Mirko H.","family":"Wagner","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,2,14]]},"reference":[{"key":"1_CR1","unstructured":"Webpage with instances, results, proofs, and the independent proof verifier (2025). https:\/\/tcs.uos.de\/research\/cr"},{"issue":"5\u20136","key":"1_CR2","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1016\/S0925-7721(96)00005-3","volume":"7","author":"G Di Battista","year":"1997","unstructured":"Di Battista, G., Garg, A., Liotta, G., Tamassia, R., Tassinari, E., Vargiu, F.: An experimental comparison of four graph drawing algorithms. Comput. Geom. 7(5\u20136), 303\u2013325 (1997)","journal-title":"Comput. Geom."},{"key":"1_CR3","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1007\/s00453-016-0200-5","volume":"79","author":"MA Bekos","year":"2017","unstructured":"Bekos, M.A., Cornelsen, S., Grilli, L., Hong, S., Kaufmann, M.: On the recognition of fan-planar and maximal outer-fan-planar graphs. Algorithmica 79, 401\u2013427 (2017)","journal-title":"Algorithmica"},{"issue":"2","key":"1_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.7155\/jgaa.v28i2.2925","volume":"28","author":"C Binucci","year":"2024","unstructured":"Binucci, C., et al.: Min-k-planar drawings of graphs. J. Graph Algorithms Appl. 28(2), 1\u201335 (2024)","journal-title":"J. Graph Algorithms Appl."},{"key":"1_CR5","doi-asserted-by":"publisher","first-page":"76","DOI":"10.1016\/j.tcs.2015.04.020","volume":"589","author":"C Binucci","year":"2015","unstructured":"Binucci, C., et al.: Fan-planarity: properties and complexity. Theor. Comput. Sci. 589, 76\u201386 (2015)","journal-title":"Theor. Comput. Sci."},{"issue":"3","key":"1_CR6","doi-asserted-by":"publisher","first-page":"241","DOI":"10.7155\/jgaa.00091","volume":"8","author":"JM Boyer","year":"2004","unstructured":"Boyer, J.M., Myrvold, W.J.: On the cutting edge: simplified o(n) planarity by edge addition. J. Graph Algorithms Appl. 8(3), 241\u2013273 (2004)","journal-title":"J. Graph Algorithms Appl."},{"issue":"2","key":"1_CR7","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1016\/j.disopt.2007.05.006","volume":"5","author":"C Buchheim","year":"2008","unstructured":"Buchheim, C., et al.: A branch-and-cut approach to the crossing number problem. Discret. Optim. 5(2), 373\u2013388 (2008)","journal-title":"Discret. Optim."},{"issue":"1","key":"1_CR8","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1137\/09076965X","volume":"25","author":"M Chimani","year":"2011","unstructured":"Chimani, M.: Facets in the crossing number polytope. SIAM J. Discret. Math. 25(1), 95\u2013111 (2011)","journal-title":"SIAM J. Discret. Math."},{"key":"1_CR9","unstructured":"Chimani, M.,\u00a0Donzelmann, N.,\u00a0Kloster, T.,\u00a0Koch, M.,\u00a0V\u00f6llering, J., Wagner, M.H.: Crossing numbers of beyond planar graphs re-revisited: a framework approach. In: Proc. GD 2024, LIPIcs 320, pp. 1\u201317 (2024)"},{"issue":"7","key":"1_CR10","doi-asserted-by":"publisher","first-page":"1838","DOI":"10.1016\/j.disc.2007.12.078","volume":"309","author":"M Chimani","year":"2009","unstructured":"Chimani, M., Gutwenger, C.: Non-planar core reduction of graphs. Discret. Math. 309(7), 1838\u20131855 (2009)","journal-title":"Discret. Math."},{"issue":"3","key":"1_CR11","doi-asserted-by":"publisher","first-page":"729","DOI":"10.7155\/jgaa.00264","volume":"16","author":"M Chimani","year":"2012","unstructured":"Chimani, M., Gutwenger, C.: Advances in the planarization method: effective multiple edge insertions. J. Graph Algorithms Appl. 16(3), 729\u2013757 (2012)","journal-title":"J. Graph Algorithms Appl."},{"key":"1_CR12","unstructured":"Chimani, M.,\u00a0Gutwenger, C.,\u00a0J\u00fcnger, M., Klau, G.W.,\u00a0Klein, K.,\u00a0Mutzel, P.: The open graph drawing framework (OGDF). In: Handbook on Graph Drawing and Visualization, pp. 543\u2013569. Chapman and Hall\/CRC (2013). http:\/\/www.ogdf.net\/"},{"key":"1_CR13","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1145\/1498698.1564504","volume":"14","author":"M Chimani","year":"2009","unstructured":"Chimani, M., Gutwenger, C., Mutzel, P.: Experiments on exact crossing minimization using column generation. ACM J. Exp. Algorithmics 14, 3\u20134 (2009)","journal-title":"ACM J. Exp. Algorithmics"},{"key":"1_CR14","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1007\/978-3-030-92931-2_3","volume-title":"Graph Drawing and Network Visualization","author":"M Chimani","year":"2021","unstructured":"Chimani, M., Ilsen, M., Wiedera, T.: Star-struck by fixed embeddings: modern crossing number heuristics. In: Purchase, H.C., Rutter, I. (eds.) GD 2021. LNCS, vol. 12868, pp. 41\u201356. Springer, Cham (2021). https:\/\/doi.org\/10.1007\/978-3-030-92931-2_3"},{"key":"1_CR15","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1016\/j.tcs.2021.10.016","volume":"898","author":"M Chimani","year":"2022","unstructured":"Chimani, M., Kindermann, P., Montecchiani, F., Valtr, P.: Crossing numbers of beyond-planar graphs. Theor. Comput. Sci. 898, 44\u201349 (2022)","journal-title":"Theor. Comput. Sci."},{"key":"1_CR16","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"284","DOI":"10.1007\/978-3-540-87744-8_24","volume-title":"Algorithms - ESA 2008","author":"M Chimani","year":"2008","unstructured":"Chimani, M., Mutzel, P., Bomze, I.: A new approach to exact crossing minimization. In: Halperin, D., Mehlhorn, K. (eds.) ESA 2008. LNCS, vol. 5193, pp. 284\u2013296. Springer, Heidelberg (2008). https:\/\/doi.org\/10.1007\/978-3-540-87744-8_24"},{"key":"1_CR17","doi-asserted-by":"crossref","unstructured":"Chimani, M.,\u00a0Mutzel, P., Schmidt, J.M.: Efficient extraction of multiple Kuratowski subdivisions. In: Proc. GD 2007, LNCS 4875, pp. 159\u2013170 (2007)","DOI":"10.1007\/978-3-540-77537-9_17"},{"key":"1_CR18","unstructured":"Chimani, M.,\u00a0Wiedera, T.: Crossing number web compute (2016). http:\/\/crossings.uos.de"},{"key":"1_CR19","unstructured":"Chimani, M.,\u00a0Wiedera, T.: An ILP-based proof system for the crossing number problem. In: Proc. ESA 2016, LIPIcs 57, pp. 1\u201313 (2016)"},{"issue":"5","key":"1_CR20","doi-asserted-by":"publisher","first-page":"1017","DOI":"10.1142\/S0129054106004248","volume":"17","author":"H de Fraysseix","year":"2006","unstructured":"de Fraysseix, H., Ossona de Mendez, P., Rosenstiehl, P.: Tr\u00e9maux trees and planarity. Int. J. Found. Comput. Sci. 17(5), 1017\u20131030 (2006)","journal-title":"Int. J. Found. Comput. Sci."},{"key":"1_CR21","doi-asserted-by":"crossref","unstructured":"Didimo, W.,\u00a0Liotta, G.,\u00a0Montecchiani, F.: A survey on graph drawing beyond planarity. ACM Comput. Surv. 52(1), 1\u201337 (2019)","DOI":"10.1145\/3301281"},{"issue":"3","key":"1_CR22","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1137\/0604033","volume":"4","author":"MR Garey","year":"1983","unstructured":"Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM J. Algebraic Disc. Meth. 4(3), 312\u2013316 (1983)","journal-title":"SIAM J. Algebraic Disc. Meth."},{"issue":"1","key":"1_CR23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00453-007-0010-x","volume":"49","author":"A Grigoriev","year":"2007","unstructured":"Grigoriev, A., Bodlaender, H.L.: Algorithms for graphs embeddable with few crossings per edge. Algorithmica 49(1), 1\u201311 (2007)","journal-title":"Algorithmica"},{"key":"1_CR24","doi-asserted-by":"crossref","unstructured":"Hong, S., Tokuyama, T. (ed.): Beyond Planar Graphs. Communications of NII Shonan Meetings. Springer (2020)","DOI":"10.1007\/978-981-15-6533-5"},{"issue":"4","key":"1_CR25","doi-asserted-by":"publisher","first-page":"549","DOI":"10.1145\/321850.321852","volume":"21","author":"JE Hopcroft","year":"1974","unstructured":"Hopcroft, J.E., Tarjan, R.E.: Efficient planarity testing. J. ACM 21(4), 549\u2013568 (1974)","journal-title":"J. ACM"},{"issue":"11","key":"1_CR26","doi-asserted-by":"publisher","first-page":"1325","DOI":"10.1002\/1097-024X(200009)30:11<1325::AID-SPE342>3.0.CO;2-T","volume":"30","author":"M J\u00fcnger","year":"2000","unstructured":"J\u00fcnger, M., Thienel, S.: The ABACUS system for branch-and-cut-and-price algorithms in integer programming and combinatorial optimization. Softw. Pract. Exp. 30(11), 1325\u20131352 (2000)","journal-title":"Softw. Pract. Exp."},{"key":"1_CR27","unstructured":"Katheder, J.,\u00a0Kindermann, P.,\u00a0Klute, F.,\u00a0Parada, I.,\u00a0Rutter, I.: On k-plane insertion into plane drawings. In: Proc. GD 2024, LIPIcs 320, pp. 1\u201311 (2024)"},{"issue":"1","key":"1_CR28","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1002\/jgt.21630","volume":"72","author":"VP Korzhik","year":"2013","unstructured":"Korzhik, V.P., Mohar, B.: Minimal obstructions for 1-immersions and hardness of 1-planarity testing. J. Graph Theory 72(1), 30\u201371 (2013)","journal-title":"J. Graph Theory"},{"key":"1_CR29","doi-asserted-by":"crossref","unstructured":"Kratochv\u00edl, J.: String graphs. II. recognizing string graphs is NP-hard. J. Comb. Theory B 52(1), 67\u201378 (1991)","DOI":"10.1016\/0095-8956(91)90091-W"},{"issue":"1","key":"1_CR30","doi-asserted-by":"publisher","first-page":"271","DOI":"10.4064\/fm-15-1-271-283","volume":"15","author":"K Kuratowski","year":"1930","unstructured":"Kuratowski, K.: Sur le probleme des courbes gauches en topologie. Fundam. Math. 15(1), 271\u2013283 (1930)","journal-title":"Fundam. Math."},{"key":"1_CR31","unstructured":"Da Lozzo, G.,\u00a0Didimo, W.,\u00a0Montecchiani, F.,\u00a0M\u00fcnch, M.,\u00a0Patrignani, M.,\u00a0Rutter, I.: Simple realizability of abstract topological graphs. In: Proc. ISAAC 2024, LIPIcs 322, pp. 1\u201315 (2024)"},{"key":"1_CR32","unstructured":"M\u00fcnch, M.,\u00a0Rutter, I.: Parameterized algorithms for beyond-planar crossing numbers. In: Proc. GD 2024, LIPIcs 320, pp. 1\u201316 (2024)"},{"key":"1_CR33","unstructured":"M\u00fcnch, M.,\u00a0Rutter, I.: Parameterized algorithms for crossing number with forbidden topological crossing patterns. In: Proc. EuroCG 2025 (2025)"},{"issue":"1","key":"1_CR34","doi-asserted-by":"publisher","first-page":"149","DOI":"10.7155\/jgaa.00586","volume":"26","author":"N van Beusekom","year":"2022","unstructured":"van Beusekom, N., Parada, I., Speckmann, B.: Crossing numbers of beyond-planar graphs revisited. J. Graph Algorithms Appl. 26(1), 149\u2013170 (2022)","journal-title":"J. Graph Algorithms Appl."},{"key":"1_CR35","unstructured":"Wolsey, L.A.: Integer programming. Discr. Math. Optim. (1998)"}],"container-title":["Lecture Notes in Computer Science","WALCOM: Algorithms and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-981-95-7127-7_1","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T04:16:23Z","timestamp":1774325783000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-981-95-7127-7_1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026]]},"ISBN":["9789819571260","9789819571277"],"references-count":35,"URL":"https:\/\/doi.org\/10.1007\/978-981-95-7127-7_1","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026]]},"assertion":[{"value":"14 February 2026","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"WALCOM","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Conference and Workshops on Algorithms and Computation","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Perugia","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Italy","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2026","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"4 March 2026","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"6 March 2026","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"20","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"walcom2026","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/mozart.diei.unipg.it\/walcom2026","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}