--- _id: '13974' abstract: - lang: eng text: The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r−1)+1 points in Rd, one can find a partition X=X1∪⋯∪Xr of X, such that the convex hulls of the Xi, i=1,…,r, all share a common point. In this paper, we prove a trengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span ⌊n/3⌋ vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Álvarez-Rebollar et al. guarantees ⌊n/6⌋pairwise crossing triangles. Our result generalizes to a result about simplices in Rd, d≥2. acknowledgement: "Part of the research leading to this paper was done during the 16th Gremo Workshop on Open Problems (GWOP), Waltensburg, Switzerland, June 12–16, 2018. We thank Patrick Schnider for suggesting the problem, and Stefan Felsner, Malte Milatz, and Emo Welzl for fruitful discussions during the workshop. We also thank Stefan Felsner and Manfred Scheucher for finding, communicating the example from Sect. 3.3, and the kind permission to include their visualization of the point set. We thank Dömötör Pálvölgyi, the SoCG reviewers, and DCG reviewers for various helpful comments.\r\nR. Fulek gratefully acknowledges support from Austrian Science Fund (FWF), Project M2281-N35. A. Kupavskii was supported by the Advanced Postdoc.Mobility Grant no. P300P2_177839 of the Swiss National Science Foundation. Research by P. Valtr was supported by the Grant no. 18-19158 S of the Czech Science Foundation (GAČR)." article_processing_charge: No article_type: original author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: Bernd full_name: Gärtner, Bernd last_name: Gärtner - first_name: Andrey full_name: Kupavskii, Andrey last_name: Kupavskii - first_name: Pavel full_name: Valtr, Pavel last_name: Valtr - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 citation: ama: Fulek R, Gärtner B, Kupavskii A, Valtr P, Wagner U. The crossing Tverberg theorem. Discrete and Computational Geometry. 2023. doi:10.1007/s00454-023-00532-x apa: Fulek, R., Gärtner, B., Kupavskii, A., Valtr, P., & Wagner, U. (2023). The crossing Tverberg theorem. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-023-00532-x chicago: Fulek, Radoslav, Bernd Gärtner, Andrey Kupavskii, Pavel Valtr, and Uli Wagner. “The Crossing Tverberg Theorem.” Discrete and Computational Geometry. Springer Nature, 2023. https://doi.org/10.1007/s00454-023-00532-x. ieee: R. Fulek, B. Gärtner, A. Kupavskii, P. Valtr, and U. Wagner, “The crossing Tverberg theorem,” Discrete and Computational Geometry. Springer Nature, 2023. ista: Fulek R, Gärtner B, Kupavskii A, Valtr P, Wagner U. 2023. The crossing Tverberg theorem. Discrete and Computational Geometry. mla: Fulek, Radoslav, et al. “The Crossing Tverberg Theorem.” Discrete and Computational Geometry, Springer Nature, 2023, doi:10.1007/s00454-023-00532-x. short: R. Fulek, B. Gärtner, A. Kupavskii, P. Valtr, U. Wagner, Discrete and Computational Geometry (2023). date_created: 2023-08-06T22:01:12Z date_published: 2023-07-27T00:00:00Z date_updated: 2023-12-13T12:03:35Z day: '27' department: - _id: UlWa doi: 10.1007/s00454-023-00532-x external_id: arxiv: - '1812.04911' isi: - '001038546500001' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.1812.04911 month: '07' oa: 1 oa_version: Preprint project: - _id: 261FA626-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: M02281 name: Eliminating intersections in drawings of graphs publication: Discrete and Computational Geometry publication_identifier: eissn: - 1432-0444 issn: - 0179-5376 publication_status: epub_ahead publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '6647' relation: earlier_version status: public scopus_import: '1' status: public title: The crossing Tverberg theorem type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2023' ... --- _id: '11593' abstract: - lang: eng text: 'A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The Z2 -genus of a graph G is the minimum g such that G has an independently even drawing on the orientable surface of genus g. An unpublished result by Robertson and Seymour implies that for every t, every graph of sufficiently large genus contains as a minor a projective t×t grid or one of the following so-called t -Kuratowski graphs: K3,t, or t copies of K5 or K3,3 sharing at most two common vertices. We show that the Z2-genus of graphs in these families is unbounded in t; in fact, equal to their genus. Together, this implies that the genus of a graph is bounded from above by a function of its Z2-genus, solving a problem posed by Schaefer and Štefankovič, and giving an approximate version of the Hanani–Tutte theorem on orientable surfaces. We also obtain an analogous result for Euler genus and Euler Z2-genus of graphs.' acknowledgement: "We thank Zdeněk Dvořák, Xavier Goaoc, and Pavel Paták for helpful discussions. We also thank Bojan Mohar, Paul Seymour, Gelasio Salazar, Jim Geelen, and John Maharry for information about their unpublished results related to Conjecture 3.1. Finally we thank the reviewers for corrections and suggestions for improving the presentation.\r\nSupported by Austrian Science Fund (FWF): M2281-N35. Supported by project 19-04113Y of the Czech Science Foundation (GAČR), by the Czech-French collaboration project EMBEDS II (CZ: 7AMB17FR029, FR: 38087RM), and by Charles University project UNCE/SCI/004." article_processing_charge: No article_type: original author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: Jan full_name: Kynčl, Jan last_name: Kynčl citation: ama: Fulek R, Kynčl J. The Z2-Genus of Kuratowski minors. Discrete and Computational Geometry. 2022;68:425-447. doi:10.1007/s00454-022-00412-w apa: Fulek, R., & Kynčl, J. (2022). The Z2-Genus of Kuratowski minors. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-022-00412-w chicago: Fulek, Radoslav, and Jan Kynčl. “The Z2-Genus of Kuratowski Minors.” Discrete and Computational Geometry. Springer Nature, 2022. https://doi.org/10.1007/s00454-022-00412-w. ieee: R. Fulek and J. Kynčl, “The Z2-Genus of Kuratowski minors,” Discrete and Computational Geometry, vol. 68. Springer Nature, pp. 425–447, 2022. ista: Fulek R, Kynčl J. 2022. The Z2-Genus of Kuratowski minors. Discrete and Computational Geometry. 68, 425–447. mla: Fulek, Radoslav, and Jan Kynčl. “The Z2-Genus of Kuratowski Minors.” Discrete and Computational Geometry, vol. 68, Springer Nature, 2022, pp. 425–47, doi:10.1007/s00454-022-00412-w. short: R. Fulek, J. Kynčl, Discrete and Computational Geometry 68 (2022) 425–447. date_created: 2022-07-17T22:01:56Z date_published: 2022-09-01T00:00:00Z date_updated: 2023-08-14T12:43:52Z day: '01' department: - _id: UlWa doi: 10.1007/s00454-022-00412-w external_id: arxiv: - '1803.05085' isi: - '000825014500001' intvolume: ' 68' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1803.05085 month: '09' oa: 1 oa_version: Preprint page: 425-447 project: - _id: 261FA626-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: M02281 name: Eliminating intersections in drawings of graphs publication: Discrete and Computational Geometry publication_identifier: eissn: - 1432-0444 issn: - 0179-5376 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '186' relation: earlier_version status: public scopus_import: '1' status: public title: The Z2-Genus of Kuratowski minors type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 68 year: '2022' ... --- _id: '7401' abstract: - lang: eng text: 'The genus g(G) of a graph G is the minimum g such that G has an embedding on the orientable surface M_g of genus g. A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The Z_2-genus of a graph G, denoted by g_0(G), is the minimum g such that G has an independently even drawing on M_g. By a result of Battle, Harary, Kodama and Youngs from 1962, the graph genus is additive over 2-connected blocks. In 2013, Schaefer and Stefankovic proved that the Z_2-genus of a graph is additive over 2-connected blocks as well, and asked whether this result can be extended to so-called 2-amalgamations, as an analogue of results by Decker, Glover, Huneke, and Stahl for the genus. We give the following partial answer. If G=G_1 cup G_2, G_1 and G_2 intersect in two vertices u and v, and G-u-v has k connected components (among which we count the edge uv if present), then |g_0(G)-(g_0(G_1)+g_0(G_2))|<=k+1. For complete bipartite graphs K_{m,n}, with n >= m >= 3, we prove that g_0(K_{m,n})/g(K_{m,n})=1-O(1/n). Similar results are proved also for the Euler Z_2-genus. We express the Z_2-genus of a graph using the minimum rank of partial symmetric matrices over Z_2; a problem that might be of independent interest. ' alternative_title: - LIPIcs article_number: '39' article_processing_charge: No author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: Jan full_name: Kyncl, Jan last_name: Kyncl citation: ama: 'Fulek R, Kyncl J. Z_2-Genus of graphs and minimum rank of partial symmetric matrices. In: 35th International Symposium on Computational Geometry (SoCG 2019). Vol 129. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2019. doi:10.4230/LIPICS.SOCG.2019.39' apa: 'Fulek, R., & Kyncl, J. (2019). Z_2-Genus of graphs and minimum rank of partial symmetric matrices. In 35th International Symposium on Computational Geometry (SoCG 2019) (Vol. 129). Portland, OR, United States: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2019.39' chicago: Fulek, Radoslav, and Jan Kyncl. “Z_2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices.” In 35th International Symposium on Computational Geometry (SoCG 2019), Vol. 129. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. https://doi.org/10.4230/LIPICS.SOCG.2019.39. ieee: R. Fulek and J. Kyncl, “Z_2-Genus of graphs and minimum rank of partial symmetric matrices,” in 35th International Symposium on Computational Geometry (SoCG 2019), Portland, OR, United States, 2019, vol. 129. ista: 'Fulek R, Kyncl J. 2019. Z_2-Genus of graphs and minimum rank of partial symmetric matrices. 35th International Symposium on Computational Geometry (SoCG 2019). SoCG: Symposium on Computational Geometry, LIPIcs, vol. 129, 39.' mla: Fulek, Radoslav, and Jan Kyncl. “Z_2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices.” 35th International Symposium on Computational Geometry (SoCG 2019), vol. 129, 39, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019, doi:10.4230/LIPICS.SOCG.2019.39. short: R. Fulek, J. Kyncl, in:, 35th International Symposium on Computational Geometry (SoCG 2019), Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. conference: end_date: 2019-06-21 location: Portland, OR, United States name: 'SoCG: Symposium on Computational Geometry' start_date: 2019-06-18 date_created: 2020-01-29T16:17:05Z date_published: 2019-06-01T00:00:00Z date_updated: 2021-01-12T08:13:24Z day: '01' ddc: - '000' department: - _id: UlWa doi: 10.4230/LIPICS.SOCG.2019.39 external_id: arxiv: - '1903.08637' file: - access_level: open_access checksum: aac37b09118cc0ab58cf77129e691f8c content_type: application/pdf creator: dernst date_created: 2020-02-04T09:14:31Z date_updated: 2020-07-14T12:47:57Z file_id: '7445' file_name: 2019_LIPIcs_Fulek.pdf file_size: 628347 relation: main_file file_date_updated: 2020-07-14T12:47:57Z has_accepted_license: '1' intvolume: ' 129' language: - iso: eng month: '06' oa: 1 oa_version: Published Version project: - _id: 261FA626-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: M02281 name: Eliminating intersections in drawings of graphs publication: 35th International Symposium on Computational Geometry (SoCG 2019) publication_identifier: isbn: - 978-3-95977-104-7 issn: - 1868-8969 publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' scopus_import: 1 status: public title: Z_2-Genus of graphs and minimum rank of partial symmetric matrices tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 129 year: '2019' ... --- _id: '5790' abstract: - lang: eng text: The partial representation extension problem is a recently introduced generalization of the recognition problem. A circle graph is an intersection graph of chords of a circle. We study the partial representation extension problem for circle graphs, where the input consists of a graph G and a partial representation R′ giving some predrawn chords that represent an induced subgraph of G. The question is whether one can extend R′ to a representation R of the entire graph G, that is, whether one can draw the remaining chords into a partially predrawn representation to obtain a representation of G. Our main result is an O(n3) time algorithm for partial representation extension of circle graphs, where n is the number of vertices. To show this, we describe the structure of all representations of a circle graph using split decomposition. This can be of independent interest. article_processing_charge: No article_type: original author: - first_name: Steven full_name: Chaplick, Steven last_name: Chaplick - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: Pavel full_name: Klavík, Pavel last_name: Klavík citation: ama: Chaplick S, Fulek R, Klavík P. Extending partial representations of circle graphs. Journal of Graph Theory. 2019;91(4):365-394. doi:10.1002/jgt.22436 apa: Chaplick, S., Fulek, R., & Klavík, P. (2019). Extending partial representations of circle graphs. Journal of Graph Theory. Wiley. https://doi.org/10.1002/jgt.22436 chicago: Chaplick, Steven, Radoslav Fulek, and Pavel Klavík. “Extending Partial Representations of Circle Graphs.” Journal of Graph Theory. Wiley, 2019. https://doi.org/10.1002/jgt.22436. ieee: S. Chaplick, R. Fulek, and P. Klavík, “Extending partial representations of circle graphs,” Journal of Graph Theory, vol. 91, no. 4. Wiley, pp. 365–394, 2019. ista: Chaplick S, Fulek R, Klavík P. 2019. Extending partial representations of circle graphs. Journal of Graph Theory. 91(4), 365–394. mla: Chaplick, Steven, et al. “Extending Partial Representations of Circle Graphs.” Journal of Graph Theory, vol. 91, no. 4, Wiley, 2019, pp. 365–94, doi:10.1002/jgt.22436. short: S. Chaplick, R. Fulek, P. Klavík, Journal of Graph Theory 91 (2019) 365–394. date_created: 2018-12-30T22:59:15Z date_published: 2019-08-01T00:00:00Z date_updated: 2023-08-24T14:30:43Z day: '01' department: - _id: UlWa doi: 10.1002/jgt.22436 ec_funded: 1 external_id: arxiv: - '1309.2399' isi: - '000485392800004' intvolume: ' 91' isi: 1 issue: '4' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1309.2399 month: '08' oa: 1 oa_version: Preprint page: 365-394 project: - _id: 25681D80-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '291734' name: International IST Postdoc Fellowship Programme publication: Journal of Graph Theory publication_identifier: issn: - '03649024' publication_status: published publisher: Wiley quality_controlled: '1' scopus_import: '1' status: public title: Extending partial representations of circle graphs type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 91 year: '2019' ... --- _id: '5857' abstract: - lang: eng text: 'A thrackle is a graph drawn in the plane so that every pair of its edges meet exactly once: either at a common end vertex or in a proper crossing. We prove that any thrackle of n vertices has at most 1.3984n edges. Quasi-thrackles are defined similarly, except that every pair of edges that do not share a vertex are allowed to cross an odd number of times. It is also shown that the maximum number of edges of a quasi-thrackle on n vertices is [Formula presented](n−1), and that this bound is best possible for infinitely many values of n.' article_processing_charge: No article_type: original author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: János full_name: Pach, János last_name: Pach citation: ama: 'Fulek R, Pach J. Thrackles: An improved upper bound. Discrete Applied Mathematics. 2019;259(4):266-231. doi:10.1016/j.dam.2018.12.025' apa: 'Fulek, R., & Pach, J. (2019). Thrackles: An improved upper bound. Discrete Applied Mathematics. Elsevier. https://doi.org/10.1016/j.dam.2018.12.025' chicago: 'Fulek, Radoslav, and János Pach. “Thrackles: An Improved Upper Bound.” Discrete Applied Mathematics. Elsevier, 2019. https://doi.org/10.1016/j.dam.2018.12.025.' ieee: 'R. Fulek and J. Pach, “Thrackles: An improved upper bound,” Discrete Applied Mathematics, vol. 259, no. 4. Elsevier, pp. 266–231, 2019.' ista: 'Fulek R, Pach J. 2019. Thrackles: An improved upper bound. Discrete Applied Mathematics. 259(4), 266–231.' mla: 'Fulek, Radoslav, and János Pach. “Thrackles: An Improved Upper Bound.” Discrete Applied Mathematics, vol. 259, no. 4, Elsevier, 2019, pp. 266–231, doi:10.1016/j.dam.2018.12.025.' short: R. Fulek, J. Pach, Discrete Applied Mathematics 259 (2019) 266–231. date_created: 2019-01-20T22:59:17Z date_published: 2019-04-30T00:00:00Z date_updated: 2023-08-24T14:39:33Z day: '30' department: - _id: UlWa doi: 10.1016/j.dam.2018.12.025 external_id: arxiv: - '1708.08037' isi: - '000466061100020' intvolume: ' 259' isi: 1 issue: '4' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1708.08037 month: '04' oa: 1 oa_version: Preprint page: 266-231 project: - _id: 261FA626-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: M02281 name: Eliminating intersections in drawings of graphs publication: Discrete Applied Mathematics publication_identifier: issn: - 0166218X publication_status: published publisher: Elsevier quality_controlled: '1' related_material: record: - id: '433' relation: earlier_version status: public scopus_import: '1' status: public title: 'Thrackles: An improved upper bound' type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 259 year: '2019' ... --- _id: '7034' abstract: - lang: eng text: We find a graph of genus 5 and its drawing on the orientable surface of genus 4 with every pair of independent edges crossing an even number of times. This shows that the strong Hanani–Tutte theorem cannot be extended to the orientable surface of genus 4. As a base step in the construction we use a counterexample to an extension of the unified Hanani–Tutte theorem on the torus. article_processing_charge: No article_type: original author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: Jan full_name: Kynčl, Jan last_name: Kynčl citation: ama: Fulek R, Kynčl J. Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4. Combinatorica. 2019;39(6):1267-1279. doi:10.1007/s00493-019-3905-7 apa: Fulek, R., & Kynčl, J. (2019). Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4. Combinatorica. Springer Nature. https://doi.org/10.1007/s00493-019-3905-7 chicago: Fulek, Radoslav, and Jan Kynčl. “Counterexample to an Extension of the Hanani-Tutte Theorem on the Surface of Genus 4.” Combinatorica. Springer Nature, 2019. https://doi.org/10.1007/s00493-019-3905-7. ieee: R. Fulek and J. Kynčl, “Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4,” Combinatorica, vol. 39, no. 6. Springer Nature, pp. 1267–1279, 2019. ista: Fulek R, Kynčl J. 2019. Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4. Combinatorica. 39(6), 1267–1279. mla: Fulek, Radoslav, and Jan Kynčl. “Counterexample to an Extension of the Hanani-Tutte Theorem on the Surface of Genus 4.” Combinatorica, vol. 39, no. 6, Springer Nature, 2019, pp. 1267–79, doi:10.1007/s00493-019-3905-7. short: R. Fulek, J. Kynčl, Combinatorica 39 (2019) 1267–1279. date_created: 2019-11-18T14:29:50Z date_published: 2019-10-29T00:00:00Z date_updated: 2023-08-30T07:26:25Z day: '29' department: - _id: UlWa doi: 10.1007/s00493-019-3905-7 ec_funded: 1 external_id: arxiv: - '1709.00508' isi: - '000493267200003' intvolume: ' 39' isi: 1 issue: '6' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1709.00508 month: '10' oa: 1 oa_version: Preprint page: 1267-1279 project: - _id: 25681D80-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '291734' name: International IST Postdoc Fellowship Programme - _id: 261FA626-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: M02281 name: Eliminating intersections in drawings of graphs publication: Combinatorica publication_identifier: eissn: - 1439-6912 issn: - 0209-9683 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4 type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 39 year: '2019' ... --- _id: '6982' abstract: - lang: eng text: "We present an efficient algorithm for a problem in the interface between clustering and graph embeddings. An embedding ϕ : G → M of a graph G into a 2-manifold M maps the vertices in V(G) to distinct points and the edges in E(G) to interior-disjoint Jordan arcs between the corresponding vertices. In applications in clustering, cartography, and visualization, nearby vertices and edges are often bundled to the same point or overlapping arcs due to data compression or low resolution. This raises the computational problem of deciding whether a given map ϕ : G → M comes from an embedding. A map ϕ : G → M is a weak embedding if it can be perturbed into an embedding ψ ϵ : G → M with ‖ ϕ − ψ ϵ ‖ < ϵ for every ϵ > 0, where ‖.‖ is the unform norm.\r\nA polynomial-time algorithm for recognizing weak embeddings has recently been found by Fulek and Kynčl. It reduces the problem to solving a system of linear equations over Z2. It runs in O(n2ω)≤ O(n4.75) time, where ω ∈ [2,2.373) is the matrix multiplication exponent and n is the number of vertices and edges of G. We improve the running time to O(n log n). Our algorithm is also conceptually simpler: We perform a sequence of local operations that gradually “untangles” the image ϕ(G) into an embedding ψ(G) or reports that ϕ is not a weak embedding. It combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations.\r\n" article_number: '50' article_type: original author: - first_name: Hugo full_name: Akitaya, Hugo last_name: Akitaya - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: Csaba full_name: Tóth, Csaba last_name: Tóth citation: ama: Akitaya H, Fulek R, Tóth C. Recognizing weak embeddings of graphs. ACM Transactions on Algorithms. 2019;15(4). doi:10.1145/3344549 apa: Akitaya, H., Fulek, R., & Tóth, C. (2019). Recognizing weak embeddings of graphs. ACM Transactions on Algorithms. ACM. https://doi.org/10.1145/3344549 chicago: Akitaya, Hugo, Radoslav Fulek, and Csaba Tóth. “Recognizing Weak Embeddings of Graphs.” ACM Transactions on Algorithms. ACM, 2019. https://doi.org/10.1145/3344549. ieee: H. Akitaya, R. Fulek, and C. Tóth, “Recognizing weak embeddings of graphs,” ACM Transactions on Algorithms, vol. 15, no. 4. ACM, 2019. ista: Akitaya H, Fulek R, Tóth C. 2019. Recognizing weak embeddings of graphs. ACM Transactions on Algorithms. 15(4), 50. mla: Akitaya, Hugo, et al. “Recognizing Weak Embeddings of Graphs.” ACM Transactions on Algorithms, vol. 15, no. 4, 50, ACM, 2019, doi:10.1145/3344549. short: H. Akitaya, R. Fulek, C. Tóth, ACM Transactions on Algorithms 15 (2019). date_created: 2019-11-04T15:45:17Z date_published: 2019-10-01T00:00:00Z date_updated: 2023-09-15T12:19:31Z day: '01' department: - _id: UlWa doi: 10.1145/3344549 external_id: arxiv: - '1709.09209' intvolume: ' 15' issue: '4' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1709.09209 month: '10' oa: 1 oa_version: Preprint project: - _id: 261FA626-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: M02281 name: Eliminating intersections in drawings of graphs publication: ACM Transactions on Algorithms publication_status: published publisher: ACM quality_controlled: '1' related_material: record: - id: '309' relation: earlier_version status: public scopus_import: 1 status: public title: Recognizing weak embeddings of graphs type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 15 year: '2019' ... --- _id: '6647' abstract: - lang: eng text: The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r-1)+1 points in R^d, one can find a partition X=X_1 cup ... cup X_r of X, such that the convex hulls of the X_i, i=1,...,r, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span floor[n/3] vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Alvarez-Rebollar et al. guarantees floor[n/6] pairwise crossing triangles. Our result generalizes to a result about simplices in R^d,d >=2. alternative_title: - LIPIcs author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: Bernd full_name: Gärtner, Bernd last_name: Gärtner - first_name: Andrey full_name: Kupavskii, Andrey last_name: Kupavskii - first_name: Pavel full_name: Valtr, Pavel last_name: Valtr - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 citation: ama: 'Fulek R, Gärtner B, Kupavskii A, Valtr P, Wagner U. The crossing Tverberg theorem. In: 35th International Symposium on Computational Geometry. Vol 129. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2019:38:1-38:13. doi:10.4230/LIPICS.SOCG.2019.38' apa: 'Fulek, R., Gärtner, B., Kupavskii, A., Valtr, P., & Wagner, U. (2019). The crossing Tverberg theorem. In 35th International Symposium on Computational Geometry (Vol. 129, p. 38:1-38:13). Portland, OR, United States: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2019.38' chicago: Fulek, Radoslav, Bernd Gärtner, Andrey Kupavskii, Pavel Valtr, and Uli Wagner. “The Crossing Tverberg Theorem.” In 35th International Symposium on Computational Geometry, 129:38:1-38:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. https://doi.org/10.4230/LIPICS.SOCG.2019.38. ieee: R. Fulek, B. Gärtner, A. Kupavskii, P. Valtr, and U. Wagner, “The crossing Tverberg theorem,” in 35th International Symposium on Computational Geometry, Portland, OR, United States, 2019, vol. 129, p. 38:1-38:13. ista: 'Fulek R, Gärtner B, Kupavskii A, Valtr P, Wagner U. 2019. The crossing Tverberg theorem. 35th International Symposium on Computational Geometry. SoCG 2019: Symposium on Computational Geometry, LIPIcs, vol. 129, 38:1-38:13.' mla: Fulek, Radoslav, et al. “The Crossing Tverberg Theorem.” 35th International Symposium on Computational Geometry, vol. 129, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019, p. 38:1-38:13, doi:10.4230/LIPICS.SOCG.2019.38. short: R. Fulek, B. Gärtner, A. Kupavskii, P. Valtr, U. Wagner, in:, 35th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019, p. 38:1-38:13. conference: end_date: 2019-06-21 location: Portland, OR, United States name: 'SoCG 2019: Symposium on Computational Geometry' start_date: 2019-06-18 date_created: 2019-07-17T10:35:04Z date_published: 2019-06-01T00:00:00Z date_updated: 2023-12-13T12:03:35Z day: '01' ddc: - '000' - '510' department: - _id: UlWa doi: 10.4230/LIPICS.SOCG.2019.38 external_id: arxiv: - '1812.04911' file: - access_level: open_access checksum: d6d017f8b41291b94d102294fa96ae9c content_type: application/pdf creator: dernst date_created: 2019-07-24T06:54:52Z date_updated: 2020-07-14T12:47:35Z file_id: '6667' file_name: 2019_LIPICS_Fulek.pdf file_size: 559837 relation: main_file file_date_updated: 2020-07-14T12:47:35Z has_accepted_license: '1' intvolume: ' 129' language: - iso: eng month: '06' oa: 1 oa_version: Published Version page: 38:1-38:13 project: - _id: 261FA626-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: M02281 name: Eliminating intersections in drawings of graphs publication: 35th International Symposium on Computational Geometry publication_identifier: isbn: - '9783959771047' issn: - 1868-8969 publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' related_material: record: - id: '13974' relation: later_version status: public scopus_import: 1 status: public title: The crossing Tverberg theorem tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 129 year: '2019' ... --- _id: '185' abstract: - lang: eng text: We resolve in the affirmative conjectures of A. Skopenkov and Repovš (1998), and M. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our proof of this result is constructive and almost immediately implies an efficient algorithm for testing whether a given piecewise linear map of a graph in a surface is approximable by an embedding. More precisely, an instance of this problem consists of (i) a graph G whose vertices are partitioned into clusters and whose inter-cluster edges are partitioned into bundles, and (ii) a region R of a 2-dimensional compact surface M given as the union of a set of pairwise disjoint discs corresponding to the clusters and a set of pairwise disjoint "pipes" corresponding to the bundles, connecting certain pairs of these discs. We are to decide whether G can be embedded inside M so that the vertices in every cluster are drawn in the corresponding disc, the edges in every bundle pass only through its corresponding pipe, and every edge crosses the boundary of each disc at most once. alternative_title: - Leibniz International Proceedings in Information, LIPIcs article_number: '39' author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: Jan full_name: Kynčl, Jan last_name: Kynčl citation: ama: 'Fulek R, Kynčl J. Hanani-Tutte for approximating maps of graphs. In: Vol 99. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2018. doi:10.4230/LIPIcs.SoCG.2018.39' apa: 'Fulek, R., & Kynčl, J. (2018). Hanani-Tutte for approximating maps of graphs (Vol. 99). Presented at the SoCG: Symposium on Computational Geometry, Budapest, Hungary: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2018.39' chicago: Fulek, Radoslav, and Jan Kynčl. “Hanani-Tutte for Approximating Maps of Graphs,” Vol. 99. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. https://doi.org/10.4230/LIPIcs.SoCG.2018.39. ieee: 'R. Fulek and J. Kynčl, “Hanani-Tutte for approximating maps of graphs,” presented at the SoCG: Symposium on Computational Geometry, Budapest, Hungary, 2018, vol. 99.' ista: 'Fulek R, Kynčl J. 2018. Hanani-Tutte for approximating maps of graphs. SoCG: Symposium on Computational Geometry, Leibniz International Proceedings in Information, LIPIcs, vol. 99, 39.' mla: Fulek, Radoslav, and Jan Kynčl. Hanani-Tutte for Approximating Maps of Graphs. Vol. 99, 39, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018, doi:10.4230/LIPIcs.SoCG.2018.39. short: R. Fulek, J. Kynčl, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. conference: end_date: 2018-06-14 location: Budapest, Hungary name: 'SoCG: Symposium on Computational Geometry' start_date: 2018-06-11 date_created: 2018-12-11T11:45:04Z date_published: 2018-01-01T00:00:00Z date_updated: 2021-01-12T06:53:36Z day: '01' ddc: - '510' department: - _id: UlWa doi: 10.4230/LIPIcs.SoCG.2018.39 file: - access_level: open_access checksum: f1b94f1a75b37c414a1f61d59fb2cd4c content_type: application/pdf creator: dernst date_created: 2018-12-17T12:33:52Z date_updated: 2020-07-14T12:45:19Z file_id: '5701' file_name: 2018_LIPIcs_Fulek.pdf file_size: 718857 relation: main_file file_date_updated: 2020-07-14T12:45:19Z has_accepted_license: '1' intvolume: ' 99' language: - iso: eng month: '01' oa: 1 oa_version: Published Version project: - _id: 261FA626-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: M02281 name: Eliminating intersections in drawings of graphs publication_identifier: isbn: - 978-3-95977-066-8 publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik publist_id: '7735' quality_controlled: '1' scopus_import: 1 status: public title: Hanani-Tutte for approximating maps of graphs tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 99 year: '2018' ... --- _id: '186' abstract: - lang: eng text: 'A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The ℤ2-genus of a graph G is the minimum g such that G has an independently even drawing on the orientable surface of genus g. An unpublished result by Robertson and Seymour implies that for every t, every graph of sufficiently large genus contains as a minor a projective t × t grid or one of the following so-called t-Kuratowski graphs: K3, t, or t copies of K5 or K3,3 sharing at most 2 common vertices. We show that the ℤ2-genus of graphs in these families is unbounded in t; in fact, equal to their genus. Together, this implies that the genus of a graph is bounded from above by a function of its ℤ2-genus, solving a problem posed by Schaefer and Štefankovič, and giving an approximate version of the Hanani-Tutte theorem on orientable surfaces.' alternative_title: - LIPIcs article_processing_charge: No author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: Jan full_name: Kynčl, Jan last_name: Kynčl citation: ama: 'Fulek R, Kynčl J. The ℤ2-Genus of Kuratowski minors. In: Vol 99. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2018:40.1-40.14. doi:10.4230/LIPIcs.SoCG.2018.40' apa: 'Fulek, R., & Kynčl, J. (2018). The ℤ2-Genus of Kuratowski minors (Vol. 99, p. 40.1-40.14). Presented at the SoCG: Symposium on Computational Geometry, Budapest, Hungary: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2018.40' chicago: Fulek, Radoslav, and Jan Kynčl. “The ℤ2-Genus of Kuratowski Minors,” 99:40.1-40.14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. https://doi.org/10.4230/LIPIcs.SoCG.2018.40. ieee: 'R. Fulek and J. Kynčl, “The ℤ2-Genus of Kuratowski minors,” presented at the SoCG: Symposium on Computational Geometry, Budapest, Hungary, 2018, vol. 99, p. 40.1-40.14.' ista: 'Fulek R, Kynčl J. 2018. The ℤ2-Genus of Kuratowski minors. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 99, 40.1-40.14.' mla: Fulek, Radoslav, and Jan Kynčl. The ℤ2-Genus of Kuratowski Minors. Vol. 99, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018, p. 40.1-40.14, doi:10.4230/LIPIcs.SoCG.2018.40. short: R. Fulek, J. Kynčl, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018, p. 40.1-40.14. conference: end_date: 2018-06-14 location: Budapest, Hungary name: 'SoCG: Symposium on Computational Geometry' start_date: 2018-06-11 date_created: 2018-12-11T11:45:05Z date_published: 2018-06-11T00:00:00Z date_updated: 2023-08-14T12:43:51Z day: '11' department: - _id: UlWa doi: 10.4230/LIPIcs.SoCG.2018.40 external_id: arxiv: - '1803.05085' intvolume: ' 99' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1803.05085 month: '06' oa: 1 oa_version: Submitted Version page: 40.1 - 40.14 project: - _id: 261FA626-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: M02281 name: Eliminating intersections in drawings of graphs publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik publist_id: '7734' quality_controlled: '1' related_material: record: - id: '11593' relation: later_version status: public scopus_import: '1' status: public title: The ℤ2-Genus of Kuratowski minors type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 99 year: '2018' ... --- _id: '433' abstract: - lang: eng text: 'A thrackle is a graph drawn in the plane so that every pair of its edges meet exactly once: either at a common end vertex or in a proper crossing. We prove that any thrackle of n vertices has at most 1.3984n edges. Quasi-thrackles are defined similarly, except that every pair of edges that do not share a vertex are allowed to cross an odd number of times. It is also shown that the maximum number of edges of a quasi-thrackle on n vertices is 3/2(n-1), and that this bound is best possible for infinitely many values of n.' alternative_title: - LNCS author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: János full_name: Pach, János last_name: Pach citation: ama: 'Fulek R, Pach J. Thrackles: An improved upper bound. In: Vol 10692. Springer; 2018:160-166. doi:10.1007/978-3-319-73915-1_14' apa: 'Fulek, R., & Pach, J. (2018). Thrackles: An improved upper bound (Vol. 10692, pp. 160–166). Presented at the GD 2017: Graph Drawing and Network Visualization, Boston, MA, United States: Springer. https://doi.org/10.1007/978-3-319-73915-1_14' chicago: 'Fulek, Radoslav, and János Pach. “Thrackles: An Improved Upper Bound,” 10692:160–66. Springer, 2018. https://doi.org/10.1007/978-3-319-73915-1_14.' ieee: 'R. Fulek and J. Pach, “Thrackles: An improved upper bound,” presented at the GD 2017: Graph Drawing and Network Visualization, Boston, MA, United States, 2018, vol. 10692, pp. 160–166.' ista: 'Fulek R, Pach J. 2018. Thrackles: An improved upper bound. GD 2017: Graph Drawing and Network Visualization, LNCS, vol. 10692, 160–166.' mla: 'Fulek, Radoslav, and János Pach. Thrackles: An Improved Upper Bound. Vol. 10692, Springer, 2018, pp. 160–66, doi:10.1007/978-3-319-73915-1_14.' short: R. Fulek, J. Pach, in:, Springer, 2018, pp. 160–166. conference: end_date: 2017-09-27 location: Boston, MA, United States name: 'GD 2017: Graph Drawing and Network Visualization' start_date: 201-09-25 date_created: 2018-12-11T11:46:27Z date_published: 2018-01-21T00:00:00Z date_updated: 2023-08-24T14:39:32Z day: '21' department: - _id: UlWa doi: 10.1007/978-3-319-73915-1_14 external_id: arxiv: - '1708.08037' intvolume: ' 10692' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1708.08037 month: '01' oa: 1 oa_version: Submitted Version page: 160 - 166 publication_status: published publisher: Springer publist_id: '7390' quality_controlled: '1' related_material: record: - id: '5857' relation: later_version status: public scopus_import: 1 status: public title: 'Thrackles: An improved upper bound' type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 10692 year: '2018' ... --- _id: '5791' abstract: - lang: eng text: Due to data compression or low resolution, nearby vertices and edges of a graph drawing may be bundled to a common node or arc. We model such a “compromised” drawing by a piecewise linear map φ:G → ℝ. We wish to perturb φ by an arbitrarily small ε>0 into a proper drawing (in which the vertices are distinct points, any two edges intersect in finitely many points, and no three edges have a common interior point) that minimizes the number of crossings. An ε-perturbation, for every ε>0, is given by a piecewise linear map (Formula Presented), where with ||·|| is the uniform norm (i.e., sup norm). We present a polynomial-time solution for this optimization problem when G is a cycle and the map φ has no spurs (i.e., no two adjacent edges are mapped to overlapping arcs). We also show that the problem becomes NP-complete (i) when G is an arbitrary graph and φ has no spurs, and (ii) when φ may have spurs and G is a cycle or a union of disjoint paths. alternative_title: - LNCS article_processing_charge: No author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: Csaba D. full_name: Tóth, Csaba D. last_name: Tóth citation: ama: 'Fulek R, Tóth CD. Crossing minimization in perturbed drawings. In: Vol 11282. Springer; 2018:229-241. doi:10.1007/978-3-030-04414-5_16' apa: 'Fulek, R., & Tóth, C. D. (2018). Crossing minimization in perturbed drawings (Vol. 11282, pp. 229–241). Presented at the Graph Drawing and Network Visualization, Barcelona, Spain: Springer. https://doi.org/10.1007/978-3-030-04414-5_16' chicago: Fulek, Radoslav, and Csaba D. Tóth. “Crossing Minimization in Perturbed Drawings,” 11282:229–41. Springer, 2018. https://doi.org/10.1007/978-3-030-04414-5_16. ieee: R. Fulek and C. D. Tóth, “Crossing minimization in perturbed drawings,” presented at the Graph Drawing and Network Visualization, Barcelona, Spain, 2018, vol. 11282, pp. 229–241. ista: Fulek R, Tóth CD. 2018. Crossing minimization in perturbed drawings. Graph Drawing and Network Visualization, LNCS, vol. 11282, 229–241. mla: Fulek, Radoslav, and Csaba D. Tóth. Crossing Minimization in Perturbed Drawings. Vol. 11282, Springer, 2018, pp. 229–41, doi:10.1007/978-3-030-04414-5_16. short: R. Fulek, C.D. Tóth, in:, Springer, 2018, pp. 229–241. conference: end_date: 2018-09-28 location: Barcelona, Spain name: Graph Drawing and Network Visualization start_date: 2018-09-26 date_created: 2018-12-30T22:59:15Z date_published: 2018-12-18T00:00:00Z date_updated: 2023-09-11T12:49:55Z day: '18' department: - _id: UlWa doi: 10.1007/978-3-030-04414-5_16 external_id: arxiv: - '1808.07608' isi: - '000672802500016' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1808.07608 month: '12' oa: 1 oa_version: Preprint page: 229-241 publication_identifier: isbn: - '9783030044138' publication_status: published publisher: Springer quality_controlled: '1' scopus_import: '1' status: public title: Crossing minimization in perturbed drawings type: conference user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: '11282 ' year: '2018' ... --- _id: '309' abstract: - lang: eng text: 'We present an efficient algorithm for a problem in the interface between clustering and graph embeddings. An embedding '' : G ! M of a graph G into a 2manifold M maps the vertices in V (G) to distinct points and the edges in E(G) to interior-disjoint Jordan arcs between the corresponding vertices. In applications in clustering, cartography, and visualization, nearby vertices and edges are often bundled to a common node or arc, due to data compression or low resolution. This raises the computational problem of deciding whether a given map '' : G ! M comes from an embedding. A map '' : G ! M is a weak embedding if it can be perturbed into an embedding ψ: G ! M with k'' "k < " for every " > 0. A polynomial-time algorithm for recognizing weak embeddings was recently found by Fulek and Kyncl [14], which reduces to solving a system of linear equations over Z2. It runs in O(n2!) O(n4:75) time, where 2:373 is the matrix multiplication exponent and n is the number of vertices and edges of G. We improve the running time to O(n log n). Our algorithm is also conceptually simpler than [14]: We perform a sequence of local operations that gradually "untangles" the image ''(G) into an embedding (G), or reports that '' is not a weak embedding. It generalizes a recent technique developed for the case that G is a cycle and the embedding is a simple polygon [1], and combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations.' acknowledgement: '∗Research supported in part by the NSF awards CCF-1422311 and CCF-1423615, and the Science Without Borders program. The second author gratefully acknowledges support from Austrian Science Fund (FWF): M2281-N35.' article_processing_charge: No author: - first_name: Hugo full_name: Akitaya, Hugo last_name: Akitaya - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: Csaba full_name: Tóth, Csaba last_name: Tóth citation: ama: 'Akitaya H, Fulek R, Tóth C. Recognizing weak embeddings of graphs. In: ACM; 2018:274-292. doi:10.1137/1.9781611975031.20' apa: 'Akitaya, H., Fulek, R., & Tóth, C. (2018). Recognizing weak embeddings of graphs (pp. 274–292). Presented at the SODA: Symposium on Discrete Algorithms, New Orleans, LA, USA: ACM. https://doi.org/10.1137/1.9781611975031.20' chicago: Akitaya, Hugo, Radoslav Fulek, and Csaba Tóth. “Recognizing Weak Embeddings of Graphs,” 274–92. ACM, 2018. https://doi.org/10.1137/1.9781611975031.20. ieee: 'H. Akitaya, R. Fulek, and C. Tóth, “Recognizing weak embeddings of graphs,” presented at the SODA: Symposium on Discrete Algorithms, New Orleans, LA, USA, 2018, pp. 274–292.' ista: 'Akitaya H, Fulek R, Tóth C. 2018. Recognizing weak embeddings of graphs. SODA: Symposium on Discrete Algorithms, 274–292.' mla: Akitaya, Hugo, et al. Recognizing Weak Embeddings of Graphs. ACM, 2018, pp. 274–92, doi:10.1137/1.9781611975031.20. short: H. Akitaya, R. Fulek, C. Tóth, in:, ACM, 2018, pp. 274–292. conference: end_date: 2018-01-10 location: New Orleans, LA, USA name: 'SODA: Symposium on Discrete Algorithms' start_date: 2018-01-07 date_created: 2018-12-11T11:45:45Z date_published: 2018-01-01T00:00:00Z date_updated: 2023-09-15T12:19:32Z day: '01' department: - _id: UlWa doi: 10.1137/1.9781611975031.20 external_id: arxiv: - '1709.09209' isi: - '000483921200021' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1709.09209 month: '01' oa: 1 oa_version: Preprint page: 274 - 292 project: - _id: 261FA626-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: M02281 name: Eliminating intersections in drawings of graphs publication_status: published publisher: ACM publist_id: '7556' quality_controlled: '1' related_material: record: - id: '6982' relation: later_version status: public scopus_import: '1' status: public title: Recognizing weak embeddings of graphs type: conference user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 year: '2018' ... --- _id: '1113' abstract: - lang: eng text: 'A drawing of a graph G is radial if the vertices of G are placed on concentric circles C 1 , . . . , C k with common center c , and edges are drawn radially : every edge intersects every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling. We show that a graph G is radial planar if G has a radial drawing in which every two edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the weak variant of the Hanani-Tutte theorem for radial planarity. This generalizes a result by Pach and Toth.' article_processing_charge: No article_type: original author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: Michael full_name: Pelsmajer, Michael last_name: Pelsmajer - first_name: Marcus full_name: Schaefer, Marcus last_name: Schaefer citation: ama: Fulek R, Pelsmajer M, Schaefer M. Hanani-Tutte for radial planarity. Journal of Graph Algorithms and Applications. 2017;21(1):135-154. doi:10.7155/jgaa.00408 apa: Fulek, R., Pelsmajer, M., & Schaefer, M. (2017). Hanani-Tutte for radial planarity. Journal of Graph Algorithms and Applications. Brown University. https://doi.org/10.7155/jgaa.00408 chicago: Fulek, Radoslav, Michael Pelsmajer, and Marcus Schaefer. “Hanani-Tutte for Radial Planarity.” Journal of Graph Algorithms and Applications. Brown University, 2017. https://doi.org/10.7155/jgaa.00408. ieee: R. Fulek, M. Pelsmajer, and M. Schaefer, “Hanani-Tutte for radial planarity,” Journal of Graph Algorithms and Applications, vol. 21, no. 1. Brown University, pp. 135–154, 2017. ista: Fulek R, Pelsmajer M, Schaefer M. 2017. Hanani-Tutte for radial planarity. Journal of Graph Algorithms and Applications. 21(1), 135–154. mla: Fulek, Radoslav, et al. “Hanani-Tutte for Radial Planarity.” Journal of Graph Algorithms and Applications, vol. 21, no. 1, Brown University, 2017, pp. 135–54, doi:10.7155/jgaa.00408. short: R. Fulek, M. Pelsmajer, M. Schaefer, Journal of Graph Algorithms and Applications 21 (2017) 135–154. date_created: 2018-12-11T11:50:13Z date_published: 2017-01-01T00:00:00Z date_updated: 2023-02-23T10:05:57Z day: '01' ddc: - '510' department: - _id: UlWa doi: 10.7155/jgaa.00408 ec_funded: 1 external_id: arxiv: - '1608.08662' file: - access_level: open_access content_type: application/pdf creator: dernst date_created: 2019-10-24T10:54:37Z date_updated: 2019-10-24T10:54:37Z file_id: '6967' file_name: 2017_JournalGraphAlgorithms_Fulek.pdf file_size: 573623 relation: main_file success: 1 file_date_updated: 2019-10-24T10:54:37Z has_accepted_license: '1' intvolume: ' 21' issue: '1' language: - iso: eng month: '01' oa: 1 oa_version: Published Version page: 135 - 154 project: - _id: 25681D80-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '291734' name: International IST Postdoc Fellowship Programme publication: Journal of Graph Algorithms and Applications publication_status: published publisher: Brown University publist_id: '6254' quality_controlled: '1' related_material: record: - id: '1164' relation: earlier_version status: public - id: '1595' relation: earlier_version status: public scopus_import: 1 status: public title: Hanani-Tutte for radial planarity type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 21 year: '2017' ... --- _id: '6517' abstract: - lang: eng text: A (possibly degenerate) drawing of a graph G in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation. We show that testing, whether a drawing of a planar graph G in the plane is approximable by an embedding, can be carried out in polynomial time, if a desired embedding of G belongs to a fixed isotopy class, i.e., the rotation system (or equivalently the faces) of the embedding of G and the choice of outer face are fixed. In other words, we show that c-planarity with embedded pipes is tractable for graphs with fixed embeddings. To the best of our knowledge an analogous result was previously known essentially only when G is a cycle. article_number: '34' author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 citation: ama: 'Fulek R. Embedding graphs into embedded graphs. In: Vol 92. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2017. doi:10.4230/LIPICS.ISAAC.2017.34' apa: 'Fulek, R. (2017). Embedding graphs into embedded graphs (Vol. 92). Presented at the ISAAC: International Symposium on Algorithms and Computation, Phuket, Thailand: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPICS.ISAAC.2017.34' chicago: Fulek, Radoslav. “Embedding Graphs into Embedded Graphs,” Vol. 92. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. https://doi.org/10.4230/LIPICS.ISAAC.2017.34. ieee: 'R. Fulek, “Embedding graphs into embedded graphs,” presented at the ISAAC: International Symposium on Algorithms and Computation, Phuket, Thailand, 2017, vol. 92.' ista: 'Fulek R. 2017. Embedding graphs into embedded graphs. ISAAC: International Symposium on Algorithms and Computation vol. 92, 34.' mla: Fulek, Radoslav. Embedding Graphs into Embedded Graphs. Vol. 92, 34, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017, doi:10.4230/LIPICS.ISAAC.2017.34. short: R. Fulek, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. conference: end_date: 2017-12-22 location: Phuket, Thailand name: 'ISAAC: International Symposium on Algorithms and Computation' start_date: 2017-12-09 date_created: 2019-06-04T12:11:52Z date_published: 2017-12-01T00:00:00Z date_updated: 2021-01-12T08:07:51Z day: '01' ddc: - '510' department: - _id: UlWa doi: 10.4230/LIPICS.ISAAC.2017.34 ec_funded: 1 file: - access_level: open_access checksum: fc7a643e29621c8bbe49d36b39081f31 content_type: application/pdf creator: kschuh date_created: 2019-06-04T12:20:35Z date_updated: 2020-07-14T12:47:33Z file_id: '6518' file_name: 2017_LIPIcs-Fulek.pdf file_size: 588982 relation: main_file file_date_updated: 2020-07-14T12:47:33Z has_accepted_license: '1' intvolume: ' 92' language: - iso: eng month: '12' oa: 1 oa_version: Published Version project: - _id: 25681D80-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '291734' name: International IST Postdoc Fellowship Programme - _id: 261FA626-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: M02281 name: Eliminating intersections in drawings of graphs publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' scopus_import: 1 status: public title: Embedding graphs into embedded graphs tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 92 year: '2017' ... --- _id: '795' abstract: - lang: eng text: 'We introduce a common generalization of the strong Hanani–Tutte theorem and the weak Hanani–Tutte theorem: if a graph G has a drawing D in the plane where every pair of independent edges crosses an even number of times, then G has a planar drawing preserving the rotation of each vertex whose incident edges cross each other evenly in D. The theorem is implicit in the proof of the strong Hanani–Tutte theorem by Pelsmajer, Schaefer and Štefankovič. We give a new, somewhat simpler proof.' article_number: P3.18 article_processing_charge: No article_type: original author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: Jan full_name: Kynčl, Jan last_name: Kynčl - first_name: Dömötör full_name: Pálvölgyi, Dömötör last_name: Pálvölgyi citation: ama: Fulek R, Kynčl J, Pálvölgyi D. Unified Hanani Tutte theorem. Electronic Journal of Combinatorics. 2017;24(3). doi:10.37236/6663 apa: Fulek, R., Kynčl, J., & Pálvölgyi, D. (2017). Unified Hanani Tutte theorem. Electronic Journal of Combinatorics. International Press. https://doi.org/10.37236/6663 chicago: Fulek, Radoslav, Jan Kynčl, and Dömötör Pálvölgyi. “Unified Hanani Tutte Theorem.” Electronic Journal of Combinatorics. International Press, 2017. https://doi.org/10.37236/6663. ieee: R. Fulek, J. Kynčl, and D. Pálvölgyi, “Unified Hanani Tutte theorem,” Electronic Journal of Combinatorics, vol. 24, no. 3. International Press, 2017. ista: Fulek R, Kynčl J, Pálvölgyi D. 2017. Unified Hanani Tutte theorem. Electronic Journal of Combinatorics. 24(3), P3.18. mla: Fulek, Radoslav, et al. “Unified Hanani Tutte Theorem.” Electronic Journal of Combinatorics, vol. 24, no. 3, P3.18, International Press, 2017, doi:10.37236/6663. short: R. Fulek, J. Kynčl, D. Pálvölgyi, Electronic Journal of Combinatorics 24 (2017). date_created: 2018-12-11T11:48:32Z date_published: 2017-07-28T00:00:00Z date_updated: 2022-03-18T12:58:53Z day: '28' ddc: - '000' department: - _id: UlWa doi: 10.37236/6663 ec_funded: 1 file: - access_level: open_access checksum: ef320cff0f062051e858f929be6a3581 content_type: application/pdf creator: dernst date_created: 2019-01-18T14:04:08Z date_updated: 2020-07-14T12:48:06Z file_id: '5853' file_name: 2017_ElectrCombi_Fulek.pdf file_size: 236944 relation: main_file file_date_updated: 2020-07-14T12:48:06Z has_accepted_license: '1' intvolume: ' 24' issue: '3' language: - iso: eng month: '07' oa: 1 oa_version: Published Version project: - _id: 25681D80-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '291734' name: International IST Postdoc Fellowship Programme publication: Electronic Journal of Combinatorics publication_identifier: issn: - '10778926' publication_status: published publisher: International Press publist_id: '6859' quality_controlled: '1' scopus_import: '1' status: public title: Unified Hanani Tutte theorem type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 24 year: '2017' ... --- _id: '793' abstract: - lang: eng text: 'Let P be a finite point set in the plane. A cordinary triangle in P is a subset of P consisting of three non-collinear points such that each of the three lines determined by the three points contains at most c points of P . Motivated by a question of Erdös, and answering a question of de Zeeuw, we prove that there exists a constant c > 0such that P contains a c-ordinary triangle, provided that P is not contained in the union of two lines. Furthermore, the number of c-ordinary triangles in P is Ω(| P |). ' article_processing_charge: No author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: Hossein full_name: Mojarrad, Hossein last_name: Mojarrad - first_name: Márton full_name: Naszódi, Márton last_name: Naszódi - first_name: József full_name: Solymosi, József last_name: Solymosi - first_name: Sebastian full_name: Stich, Sebastian last_name: Stich - first_name: May full_name: Szedlák, May last_name: Szedlák citation: ama: 'Fulek R, Mojarrad H, Naszódi M, Solymosi J, Stich S, Szedlák M. On the existence of ordinary triangles. Computational Geometry: Theory and Applications. 2017;66:28-31. doi:10.1016/j.comgeo.2017.07.002' apa: 'Fulek, R., Mojarrad, H., Naszódi, M., Solymosi, J., Stich, S., & Szedlák, M. (2017). On the existence of ordinary triangles. Computational Geometry: Theory and Applications. Elsevier. https://doi.org/10.1016/j.comgeo.2017.07.002' chicago: 'Fulek, Radoslav, Hossein Mojarrad, Márton Naszódi, József Solymosi, Sebastian Stich, and May Szedlák. “On the Existence of Ordinary Triangles.” Computational Geometry: Theory and Applications. Elsevier, 2017. https://doi.org/10.1016/j.comgeo.2017.07.002.' ieee: 'R. Fulek, H. Mojarrad, M. Naszódi, J. Solymosi, S. Stich, and M. Szedlák, “On the existence of ordinary triangles,” Computational Geometry: Theory and Applications, vol. 66. Elsevier, pp. 28–31, 2017.' ista: 'Fulek R, Mojarrad H, Naszódi M, Solymosi J, Stich S, Szedlák M. 2017. On the existence of ordinary triangles. Computational Geometry: Theory and Applications. 66, 28–31.' mla: 'Fulek, Radoslav, et al. “On the Existence of Ordinary Triangles.” Computational Geometry: Theory and Applications, vol. 66, Elsevier, 2017, pp. 28–31, doi:10.1016/j.comgeo.2017.07.002.' short: 'R. Fulek, H. Mojarrad, M. Naszódi, J. Solymosi, S. Stich, M. Szedlák, Computational Geometry: Theory and Applications 66 (2017) 28–31.' date_created: 2018-12-11T11:48:32Z date_published: 2017-01-01T00:00:00Z date_updated: 2023-09-27T12:15:16Z day: '01' department: - _id: UlWa doi: 10.1016/j.comgeo.2017.07.002 ec_funded: 1 external_id: isi: - '000412039700003' intvolume: ' 66' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1701.08183 month: '01' oa: 1 oa_version: Submitted Version page: 28 - 31 project: - _id: 25681D80-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '291734' name: International IST Postdoc Fellowship Programme publication: 'Computational Geometry: Theory and Applications' publication_identifier: issn: - '09257721' publication_status: published publisher: Elsevier publist_id: '6861' quality_controlled: '1' status: public title: On the existence of ordinary triangles type: journal_article user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 66 year: '2017' ... --- _id: '794' abstract: - lang: eng text: We show that c-planarity is solvable in quadratic time for flat clustered graphs with three clusters if the combinatorial embedding of the underlying graph is fixed. In simpler graph-theoretical terms our result can be viewed as follows. Given a graph G with the vertex set partitioned into three parts embedded on a 2-sphere, our algorithm decides if we can augment G by adding edges without creating an edge-crossing so that in the resulting spherical graph the vertices of each part induce a connected sub-graph. We proceed by a reduction to the problem of testing the existence of a perfect matching in planar bipartite graphs. We formulate our result in a slightly more general setting of cyclic clustered graphs, i.e., the simple graph obtained by contracting each cluster, where we disregard loops and multi-edges, is a cycle. acknowledgement: I would like to thank Jan Kynčl, Dömötör Pálvölgyi and anonymous referees for many comments and suggestions that helped to improve the presentation of the result. article_processing_charge: No author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 citation: ama: 'Fulek R. C-planarity of embedded cyclic c-graphs. Computational Geometry: Theory and Applications. 2017;66:1-13. doi:10.1016/j.comgeo.2017.06.016' apa: 'Fulek, R. (2017). C-planarity of embedded cyclic c-graphs. Computational Geometry: Theory and Applications. Elsevier. https://doi.org/10.1016/j.comgeo.2017.06.016' chicago: 'Fulek, Radoslav. “C-Planarity of Embedded Cyclic c-Graphs.” Computational Geometry: Theory and Applications. Elsevier, 2017. https://doi.org/10.1016/j.comgeo.2017.06.016.' ieee: 'R. Fulek, “C-planarity of embedded cyclic c-graphs,” Computational Geometry: Theory and Applications, vol. 66. Elsevier, pp. 1–13, 2017.' ista: 'Fulek R. 2017. C-planarity of embedded cyclic c-graphs. Computational Geometry: Theory and Applications. 66, 1–13.' mla: 'Fulek, Radoslav. “C-Planarity of Embedded Cyclic c-Graphs.” Computational Geometry: Theory and Applications, vol. 66, Elsevier, 2017, pp. 1–13, doi:10.1016/j.comgeo.2017.06.016.' short: 'R. Fulek, Computational Geometry: Theory and Applications 66 (2017) 1–13.' date_created: 2018-12-11T11:48:32Z date_published: 2017-12-01T00:00:00Z date_updated: 2023-09-27T12:14:49Z day: '01' department: - _id: UlWa doi: 10.1016/j.comgeo.2017.06.016 external_id: isi: - '000412039700001' intvolume: ' 66' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1602.01346 month: '12' oa: 1 oa_version: Preprint page: 1 - 13 publication: 'Computational Geometry: Theory and Applications' publication_status: published publisher: Elsevier publist_id: '6860' quality_controlled: '1' related_material: record: - id: '1165' relation: earlier_version status: public scopus_import: '1' status: public title: C-planarity of embedded cyclic c-graphs type: journal_article user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 66 year: '2017' ... --- _id: '1348' abstract: - lang: eng text: 'A drawing in the plane (ℝ2) of a graph G = (V,E) equipped with a function γ : V → ℕ is x-bounded if (i) x(u) < x(v) whenever γ(u) < γ(v) and (ii) γ(u) ≤ γ(w) ≤ γ(v), where uv ∈ E and γ(u) ≤ γ(v), whenever x(w) ∈ x(uv), where x(.) denotes the projection to the xaxis.We prove a characterization of isotopy classes of embeddings of connected graphs equipped with γ in the plane containing an x-bounded embedding.Then we present an efficient algorithm, which relies on our result, for testing the existence of an x-bounded embedding if the given graph is a forest.This partially answers a question raised recently by Angelini et al.and Chang et al., and proves that c-planarity testing of flat clustered graphs with three clusters is tractable when the underlying abstract graph is a forest.' alternative_title: - LNCS author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 citation: ama: 'Fulek R. Bounded embeddings of graphs in the plane. In: Vol 9843. Springer; 2016:31-42. doi:10.1007/978-3-319-44543-4_3' apa: 'Fulek, R. (2016). Bounded embeddings of graphs in the plane (Vol. 9843, pp. 31–42). Presented at the IWOCA: International Workshop on Combinatorial Algorithms, Helsinki, Finland: Springer. https://doi.org/10.1007/978-3-319-44543-4_3' chicago: Fulek, Radoslav. “Bounded Embeddings of Graphs in the Plane,” 9843:31–42. Springer, 2016. https://doi.org/10.1007/978-3-319-44543-4_3. ieee: 'R. Fulek, “Bounded embeddings of graphs in the plane,” presented at the IWOCA: International Workshop on Combinatorial Algorithms, Helsinki, Finland, 2016, vol. 9843, pp. 31–42.' ista: 'Fulek R. 2016. Bounded embeddings of graphs in the plane. IWOCA: International Workshop on Combinatorial Algorithms, LNCS, vol. 9843, 31–42.' mla: Fulek, Radoslav. Bounded Embeddings of Graphs in the Plane. Vol. 9843, Springer, 2016, pp. 31–42, doi:10.1007/978-3-319-44543-4_3. short: R. Fulek, in:, Springer, 2016, pp. 31–42. conference: end_date: 2018-08-19 location: Helsinki, Finland name: 'IWOCA: International Workshop on Combinatorial Algorithms' start_date: 2016-08-17 date_created: 2018-12-11T11:51:31Z date_published: 2016-08-09T00:00:00Z date_updated: 2021-01-12T06:50:03Z day: '09' department: - _id: UlWa doi: 10.1007/978-3-319-44543-4_3 ec_funded: 1 intvolume: ' 9843' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1610.07144 month: '08' oa: 1 oa_version: Preprint page: 31 - 42 project: - _id: 25681D80-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '291734' name: International IST Postdoc Fellowship Programme publication_status: published publisher: Springer publist_id: '5901' quality_controlled: '1' scopus_import: 1 status: public title: Bounded embeddings of graphs in the plane type: conference user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 9843 year: '2016' ... --- _id: '1164' abstract: - lang: eng text: 'A drawing of a graph G is radial if the vertices of G are placed on concentric circles C1, … , Ck with common center c, and edges are drawn radially: every edge intersects every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling. A pair of edges e and f in a graph is independent if e and f do not share a vertex. We show that a graph G is radial planar if G has a radial drawing in which every two independent edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the strong Hanani-Tutte theorem for radial planarity. This characterization yields a very simple algorithm for radial planarity testing.' alternative_title: - LNCS article_processing_charge: No author: - first_name: Radoslav full_name: Fulek, Radoslav id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87 last_name: Fulek orcid: 0000-0001-8485-1774 - first_name: Michael full_name: Pelsmajer, Michael last_name: Pelsmajer - first_name: Marcus full_name: Schaefer, Marcus last_name: Schaefer citation: ama: 'Fulek R, Pelsmajer M, Schaefer M. Hanani-Tutte for radial planarity II. In: Vol 9801. Springer; 2016:468-481. doi:10.1007/978-3-319-50106-2_36' apa: 'Fulek, R., Pelsmajer, M., & Schaefer, M. (2016). Hanani-Tutte for radial planarity II (Vol. 9801, pp. 468–481). Presented at the GD: Graph Drawing and Network Visualization, Athens, Greece: Springer. https://doi.org/10.1007/978-3-319-50106-2_36' chicago: Fulek, Radoslav, Michael Pelsmajer, and Marcus Schaefer. “Hanani-Tutte for Radial Planarity II,” 9801:468–81. Springer, 2016. https://doi.org/10.1007/978-3-319-50106-2_36. ieee: 'R. Fulek, M. Pelsmajer, and M. Schaefer, “Hanani-Tutte for radial planarity II,” presented at the GD: Graph Drawing and Network Visualization, Athens, Greece, 2016, vol. 9801, pp. 468–481.' ista: 'Fulek R, Pelsmajer M, Schaefer M. 2016. Hanani-Tutte for radial planarity II. GD: Graph Drawing and Network Visualization, LNCS, vol. 9801, 468–481.' mla: Fulek, Radoslav, et al. Hanani-Tutte for Radial Planarity II. Vol. 9801, Springer, 2016, pp. 468–81, doi:10.1007/978-3-319-50106-2_36. short: R. Fulek, M. Pelsmajer, M. Schaefer, in:, Springer, 2016, pp. 468–481. conference: end_date: 2016-09-21 location: Athens, Greece name: 'GD: Graph Drawing and Network Visualization' start_date: 2016-09-19 date_created: 2018-12-11T11:50:29Z date_published: 2016-12-08T00:00:00Z date_updated: 2023-02-23T10:05:57Z day: '08' department: - _id: UlWa doi: 10.1007/978-3-319-50106-2_36 ec_funded: 1 external_id: arxiv: - '1608.08662' intvolume: ' 9801' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1608.08662 month: '12' oa: 1 oa_version: Preprint page: 468 - 481 project: - _id: 25681D80-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '291734' name: International IST Postdoc Fellowship Programme publication_status: published publisher: Springer publist_id: '6193' quality_controlled: '1' related_material: record: - id: '1113' relation: later_version status: public - id: '1595' relation: earlier_version status: public scopus_import: 1 status: public title: Hanani-Tutte for radial planarity II type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 9801 year: '2016' ...