--- OA_place: repository OA_type: green _id: '21374' abstract: - lang: eng text: "Let . S be a set of distinct points in general position in the\r\nEuclidean plane. A plane Hamiltonian path on . S is a crossing-free geometric path such that every point of .S is a vertex of the path. It is\r\nknown that, if. S is sufficiently large, there exist three edge-disjoint plane\r\nHamiltonian paths on . S. In this paper we study an edge-constrained\r\nversion of the problem of finding Hamiltonian paths on a point set. We\r\nfirst consider the problem of finding a single plane Hamiltonian path . π\r\nwith endpoints .s, t ∈ S and constraints given by a segment . ab, where\r\n.a, b ∈ S. We consider the following scenarios: (i) .ab ∈ π; (ii) .ab π. We\r\ncharacterize those quintuples . S, a, b, s, t for which . π exists. Secondly,\r\nwe consider the problem of finding two plane Hamiltonian paths . π1, π2\r\non a set . S with constraints given by a segment . ab, where .a, b ∈ S. We\r\nconsider the following scenarios: (i) .π1 and .π2 share no edges and .ab is\r\nan edge of . π1; (ii) .π1 and .π2 share no edges and none of them includes\r\n.ab as an edge; (iii) both .π1 and .π2 include .ab as an edge and share no\r\nother edges. In all cases, we characterize those triples . S, a, b for which\r\n.π1 and .π2 exist." acknowledgement: "We thank the organizers of the HOMONOLO 2024 workshop in Nová Louka, Czech Republic, for the fruitful atmosphere where the research on this project was initiated.\r\n\r\nT. Antić, A. Džuklevski, J. Kratochvíl and M. Saumell received funding from GAČR grant 23–04949X, T.A and A.Dž were additionally supported by GAUK grant SVV–2025–260822. G. Liotta was supported in part by MUR of Italy, PRIN Project no. 2022TS4Y3N – EXPAND and PON Project ARS01_00540. J. Fiala was in part supported by GAČR grant 25-16847S." alternative_title: - LNCS article_processing_charge: No arxiv: 1 author: - first_name: Todor full_name: Antić, Todor last_name: Antić - first_name: Aleksa full_name: Džuklevski, Aleksa last_name: Džuklevski - first_name: Jiří full_name: Fiala, Jiří last_name: Fiala - first_name: Jan full_name: Kratochvíl, Jan last_name: Kratochvíl - first_name: Giuseppe full_name: Liotta, Giuseppe last_name: Liotta - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian - first_name: Maria full_name: Saumell, Maria last_name: Saumell - first_name: Johannes full_name: Zink, Johannes last_name: Zink citation: ama: 'Antić T, Džuklevski A, Fiala J, et al. Edge-constrained Hamiltonian paths on a point set. In: 51st International Conference on Current Trends in Theory and Practice of Computer Science. Vol 16448. Springer Nature; 2026:532-546. doi:10.1007/978-3-032-17801-5_39' apa: 'Antić, T., Džuklevski, A., Fiala, J., Kratochvíl, J., Liotta, G., Saghafian, M., … Zink, J. (2026). Edge-constrained Hamiltonian paths on a point set. In 51st International Conference on Current Trends in Theory and Practice of Computer Science (Vol. 16448, pp. 532–546). Krakow, Poland: Springer Nature. https://doi.org/10.1007/978-3-032-17801-5_39' chicago: Antić, Todor, Aleksa Džuklevski, Jiří Fiala, Jan Kratochvíl, Giuseppe Liotta, Morteza Saghafian, Maria Saumell, and Johannes Zink. “Edge-Constrained Hamiltonian Paths on a Point Set.” In 51st International Conference on Current Trends in Theory and Practice of Computer Science, 16448:532–46. Springer Nature, 2026. https://doi.org/10.1007/978-3-032-17801-5_39. ieee: T. Antić et al., “Edge-constrained Hamiltonian paths on a point set,” in 51st International Conference on Current Trends in Theory and Practice of Computer Science, Krakow, Poland, 2026, vol. 16448, pp. 532–546. ista: 'Antić T, Džuklevski A, Fiala J, Kratochvíl J, Liotta G, Saghafian M, Saumell M, Zink J. 2026. Edge-constrained Hamiltonian paths on a point set. 51st International Conference on Current Trends in Theory and Practice of Computer Science. SOFSEM: Conference on Current Trends in Theory and Practice of Computer Science, LNCS, vol. 16448, 532–546.' mla: Antić, Todor, et al. “Edge-Constrained Hamiltonian Paths on a Point Set.” 51st International Conference on Current Trends in Theory and Practice of Computer Science, vol. 16448, Springer Nature, 2026, pp. 532–46, doi:10.1007/978-3-032-17801-5_39. short: T. Antić, A. Džuklevski, J. Fiala, J. Kratochvíl, G. Liotta, M. Saghafian, M. Saumell, J. Zink, in:, 51st International Conference on Current Trends in Theory and Practice of Computer Science, Springer Nature, 2026, pp. 532–546. conference: end_date: 2026-02-13 location: Krakow, Poland name: 'SOFSEM: Conference on Current Trends in Theory and Practice of Computer Science' start_date: 2026-02-09 date_created: 2026-03-01T23:01:40Z date_published: 2026-02-13T00:00:00Z date_updated: 2026-03-02T08:49:20Z day: '13' department: - _id: HeEd doi: 10.1007/978-3-032-17801-5_39 external_id: arxiv: - '2511.22526' intvolume: ' 16448' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2511.22526 month: '02' oa: 1 oa_version: Preprint page: 532-546 publication: 51st International Conference on Current Trends in Theory and Practice of Computer Science publication_identifier: eissn: - 1611-3349 isbn: - '9783032178008' issn: - 0302-9743 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Edge-constrained Hamiltonian paths on a point set type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 16448 year: '2026' ... --- OA_place: repository OA_type: green _id: '21410' abstract: - lang: eng text: Given a finite set of red and blue points in R^d, the MST-ratio is defined as the total length of the Euclidean minimum spanning trees of the red points and the blue points, divided by the length of the Euclidean minimum spanning tree of their union. The MST-ratio has recently gained attention due to its direct interpretation in topological models for studying point sets with applications in spatial biology. The maximum MST-ratio of a point set is the maximum MST-ratio over all proper colorings of its points by red and blue. We prove that finding the maximum MST-ratio of a given point set is NP-hard when the dimension is part of the input. Moreover, we present a quadratic-time 3-approximation algorithm for this problem. As part of the proof, we show that in any metric space, the maximum MST-ratio is smaller than 3. Furthermore, we study the average MST-ratio over all colorings of a set of n points. We show that this average is always at least n-2/n-1, and for n random points uniformly distributed in a d-dimensional unit cube, the average tends to (math formular) in expectation as n approaches infinity. acknowledgement: "A. J. Ameli—Supported by the project COALESCE (ERC grant no. 853234).\r\nM. Saghafian—Partially supported by the European Research Council (ERC), grant no. 788183, and by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31." alternative_title: - LNCS article_processing_charge: No arxiv: 1 author: - first_name: Afrouz full_name: Jabal Ameli, Afrouz last_name: Jabal Ameli - first_name: Faezeh full_name: Motiei, Faezeh last_name: Motiei - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: 'Jabal Ameli A, Motiei F, Saghafian M. On the MST-ratio: Theoretical bounds and complexity of finding the maximum. In: 20th International Conference and Workshops on Algorithms and Computation. Vol 16444. Springer Nature; 2026:386-401. doi:10.1007/978-981-95-7127-7_26' apa: 'Jabal Ameli, A., Motiei, F., & Saghafian, M. (2026). On the MST-ratio: Theoretical bounds and complexity of finding the maximum. In 20th International Conference and Workshops on Algorithms and Computation (Vol. 16444, pp. 386–401). Perugia, Italy: Springer Nature. https://doi.org/10.1007/978-981-95-7127-7_26' chicago: 'Jabal Ameli, Afrouz, Faezeh Motiei, and Morteza Saghafian. “On the MST-Ratio: Theoretical Bounds and Complexity of Finding the Maximum.” In 20th International Conference and Workshops on Algorithms and Computation, 16444:386–401. Springer Nature, 2026. https://doi.org/10.1007/978-981-95-7127-7_26.' ieee: 'A. Jabal Ameli, F. Motiei, and M. Saghafian, “On the MST-ratio: Theoretical bounds and complexity of finding the maximum,” in 20th International Conference and Workshops on Algorithms and Computation, Perugia, Italy, 2026, vol. 16444, pp. 386–401.' ista: 'Jabal Ameli A, Motiei F, Saghafian M. 2026. On the MST-ratio: Theoretical bounds and complexity of finding the maximum. 20th International Conference and Workshops on Algorithms and Computation. WALCOM: International Conference and Workshops on Algorithms and Computation, LNCS, vol. 16444, 386–401.' mla: 'Jabal Ameli, Afrouz, et al. “On the MST-Ratio: Theoretical Bounds and Complexity of Finding the Maximum.” 20th International Conference and Workshops on Algorithms and Computation, vol. 16444, Springer Nature, 2026, pp. 386–401, doi:10.1007/978-981-95-7127-7_26.' short: A. Jabal Ameli, F. Motiei, M. Saghafian, in:, 20th International Conference and Workshops on Algorithms and Computation, Springer Nature, 2026, pp. 386–401. conference: end_date: 2026-03-06 location: Perugia, Italy name: 'WALCOM: International Conference and Workshops on Algorithms and Computation' start_date: 2026-03-04 date_created: 2026-03-08T23:01:45Z date_published: 2026-02-14T00:00:00Z date_updated: 2026-03-09T10:25:41Z day: '14' department: - _id: HeEd doi: 10.1007/978-981-95-7127-7_26 ec_funded: 1 external_id: arxiv: - '2409.11079' intvolume: ' 16444' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2409.11079 month: '02' oa: 1 oa_version: Preprint page: 386-401 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science publication: 20th International Conference and Workshops on Algorithms and Computation publication_identifier: eissn: - 1611-3349 isbn: - '9789819571260' issn: - 0302-9743 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: 'On the MST-ratio: Theoretical bounds and complexity of finding the maximum' type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 16444 year: '2026' ... --- OA_place: repository OA_type: green _id: '21781' abstract: - lang: eng text: "Given a set A of n points (vertices) in general position in the plane, the complete geometric graph \r\nKn[A] consists of all (n2) segments (edges) between the elements of A. It is known that the edge set of every complete geometric graph on n vertices can be partitioned into O(n3∕2) crossing-free paths (or matchings). We strengthen this result under various additional assumptions on the point set. In particular, we prove that for a set A of n randomly selected points, uniformly distributed in [0,1]2, with probability tending to 1 as n→∞, the edge set of Kn[A] can be covered by O(nlogn) crossing-free paths and by O(n√logn) crossing-free matchings. On the other hand, we construct n-element point sets such that covering the edge set of Kn[A] requires a quadratic number of monotone paths." acknowledgement: "Research partially supported by ERC Advanced Grant \"GeoScape\", no. 882971 and\r\nHungarian NKFIH grant no. K-131529. Work by the third author is supported by EPSRC grant\r\nEP/X013642/1. Work by the third author is partially supported by the European Research Council (ERC), grant no. 788183, and by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31." article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Adrian full_name: Dumitrescu, Adrian last_name: Dumitrescu - first_name: János full_name: Pach, János last_name: Pach - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian - first_name: Alex full_name: Scott, Alex last_name: Scott citation: ama: Dumitrescu A, Pach J, Saghafian M, Scott A. Covering complete geometric graphs by monotone paths. Combinatorics and Number Theory. 2026;15(1):73-82. doi:10.2140/cnt.2026.15.73 apa: Dumitrescu, A., Pach, J., Saghafian, M., & Scott, A. (2026). Covering complete geometric graphs by monotone paths. Combinatorics and Number Theory. Mathematical Sciences Publishers. https://doi.org/10.2140/cnt.2026.15.73 chicago: Dumitrescu, Adrian, János Pach, Morteza Saghafian, and Alex Scott. “Covering Complete Geometric Graphs by Monotone Paths.” Combinatorics and Number Theory. Mathematical Sciences Publishers, 2026. https://doi.org/10.2140/cnt.2026.15.73. ieee: A. Dumitrescu, J. Pach, M. Saghafian, and A. Scott, “Covering complete geometric graphs by monotone paths,” Combinatorics and Number Theory, vol. 15, no. 1. Mathematical Sciences Publishers, pp. 73–82, 2026. ista: Dumitrescu A, Pach J, Saghafian M, Scott A. 2026. Covering complete geometric graphs by monotone paths. Combinatorics and Number Theory. 15(1), 73–82. mla: Dumitrescu, Adrian, et al. “Covering Complete Geometric Graphs by Monotone Paths.” Combinatorics and Number Theory, vol. 15, no. 1, Mathematical Sciences Publishers, 2026, pp. 73–82, doi:10.2140/cnt.2026.15.73. short: A. Dumitrescu, J. Pach, M. Saghafian, A. Scott, Combinatorics and Number Theory 15 (2026) 73–82. date_created: 2026-05-03T22:01:37Z date_published: 2026-04-17T00:00:00Z date_updated: 2026-05-07T07:45:24Z day: '17' department: - _id: HeEd doi: 10.2140/cnt.2026.15.73 ec_funded: 1 external_id: arxiv: - '2507.10840' intvolume: ' 15' issue: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2507.10840 month: '04' oa: 1 oa_version: Preprint page: 73-82 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science publication: Combinatorics and Number Theory publication_identifier: eissn: - 2996-220X issn: - 2996-2196 publication_status: published publisher: Mathematical Sciences Publishers quality_controlled: '1' scopus_import: '1' status: public title: Covering complete geometric graphs by monotone paths type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 15 year: '2026' ... --- OA_place: publisher OA_type: hybrid PlanS_conform: '1' _id: '20456' abstract: - lang: eng text: Given a locally finite set A⊆Rd and a coloring χ:A→{0,1,…,s}, we introduce the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in Rs+d that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with n=#A, and the coloring is random, then the chromatic Delaunay mosaic has O(n⌈d/2⌉) cells in expectation. In contrast, for Delone sets and Poisson point processes in Rd, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in R2 all colorings of a dense set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications. acknowledgement: The fourth author thanks Boris Aronov for insightful discussions on the size of the overlay of Voronoi tessellations. Open access funding provided by Institute of Science and Technology (IST Austria). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35. article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. Discrete and Computational Geometry. 2026;75:24-47. doi:10.1007/s00454-025-00778-7 apa: Biswas, R., Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (2026). On the size of chromatic Delaunay mosaics. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-025-00778-7 chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “On the Size of Chromatic Delaunay Mosaics.” Discrete and Computational Geometry. Springer Nature, 2026. https://doi.org/10.1007/s00454-025-00778-7. ieee: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “On the size of chromatic Delaunay mosaics,” Discrete and Computational Geometry, vol. 75. Springer Nature, pp. 24–47, 2026. ista: Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2026. On the size of chromatic Delaunay mosaics. Discrete and Computational Geometry. 75, 24–47. mla: Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” Discrete and Computational Geometry, vol. 75, Springer Nature, 2026, pp. 24–47, doi:10.1007/s00454-025-00778-7. short: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Discrete and Computational Geometry 75 (2026) 24–47. corr_author: '1' date_created: 2025-10-12T22:01:26Z date_published: 2026-01-01T00:00:00Z date_updated: 2026-01-05T13:21:56Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.1007/s00454-025-00778-7 ec_funded: 1 external_id: arxiv: - '2212.03121' isi: - '001584166900001' file: - access_level: open_access checksum: 0addb5c1b78142f9fb453bfa04695400 content_type: application/pdf creator: dernst date_created: 2026-01-05T13:21:20Z date_updated: 2026-01-05T13:21:20Z file_id: '20952' file_name: 2026_DiscreteCompGeom_Biswas.pdf file_size: 570922 relation: main_file success: 1 file_date_updated: 2026-01-05T13:21:20Z has_accepted_license: '1' intvolume: ' 75' isi: 1 language: - iso: eng month: '01' oa: 1 oa_version: Published Version page: 24-47 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes 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: '15090' relation: earlier_version status: public scopus_import: '1' status: public title: On the size of chromatic Delaunay mosaics 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: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 75 year: '2026' ... --- OA_place: repository OA_type: green _id: '20490' abstract: - lang: eng text: "We study flips in hypertriangulations of planar points sets. Here a level-k hypertriangulation of n\r\n points in the plane is a subdivision induced by the projection of a k-hypersimplex, which is the convex hull of the barycenters of the (k-1)-dimensional faces of the standard (n-1)-simplex. In particular, we introduce four types of flips and prove that the level-2 hypertriangulations are connected by these flips.\r\n" acknowledgement: Work by all authors but the second is supported by the European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. Work by the second author is partially supported by the Alexander von Humboldt Foundation and by the Simons Foundation . The second author thanks Jesús A. De Loera for useful discussions on flips and non-flips and Pavel Galashin and Alexey Balitskiy for useful discussions on plabic graphs. article_number: '104248' article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Alexey full_name: Garber, Alexey last_name: Garber - first_name: Mohadese full_name: Ghafari, Mohadese last_name: Ghafari - first_name: Teresa full_name: Heiss, Teresa id: 4879BB4E-F248-11E8-B48F-1D18A9856A87 last_name: Heiss orcid: 0000-0002-1780-2689 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. 2026;132. doi:10.1016/j.ejc.2025.104248 apa: Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., & Saghafian, M. (2026). Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. Elsevier. https://doi.org/10.1016/j.ejc.2025.104248 chicago: Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “Flips in Two-Dimensional Hypertriangulations.” European Journal of Combinatorics. Elsevier, 2026. https://doi.org/10.1016/j.ejc.2025.104248. ieee: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “Flips in two-dimensional hypertriangulations,” European Journal of Combinatorics, vol. 132. Elsevier, 2026. ista: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2026. Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. 132, 104248. mla: Edelsbrunner, Herbert, et al. “Flips in Two-Dimensional Hypertriangulations.” European Journal of Combinatorics, vol. 132, 104248, Elsevier, 2026, doi:10.1016/j.ejc.2025.104248. short: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, European Journal of Combinatorics 132 (2026). corr_author: '1' das_tickbox: '0' date_created: 2025-10-19T22:01:31Z date_published: 2026-02-01T00:00:00Z date_updated: 2026-07-23T11:58:37Z day: '01' department: - _id: HeEd doi: 10.1016/j.ejc.2025.104248 ec_funded: 1 external_id: arxiv: - '2212.11380' isi: - '001599061500002' intvolume: ' 132' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2212.11380 month: '02' oa: 1 oa_version: Preprint project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: European Journal of Combinatorics publication_identifier: issn: - 0195-6698 publication_status: published publisher: Elsevier quality_controlled: '1' scopus_import: '1' status: public title: Flips in two-dimensional hypertriangulations type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 132 year: '2026' ... --- OA_place: repository OA_type: green _id: '20585' abstract: - lang: eng text: Motivated by applications in medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on the persistent homology for images, kernels and cokernels, we design provably stable homological quantifiers that describe the geometric micro- and macro-structure of how the color classes mingle. These can be efficiently computed using chromatic variants of Delaunay and alpha complexes, and code that does these computations is provided. acknowledgement: "This project has received funding from the European Research\r\nCouncil (ERC) under the European Union’s Horizon 2020 research and innovation\r\nprogramme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund\r\n(FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR\r\n109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF),\r\ngrant no. I 02979-N35." article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. Foundations of Data Science. 2026;8:30-62. doi:10.3934/fods.2025003 apa: Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (2026). Chromatic alpha complexes. Foundations of Data Science. AIMS. https://doi.org/10.3934/fods.2025003 chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Alpha Complexes.” Foundations of Data Science. AIMS, 2026. https://doi.org/10.3934/fods.2025003. ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic alpha complexes,” Foundations of Data Science, vol. 8. AIMS, pp. 30–62, 2026. ista: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2026. Chromatic alpha complexes. Foundations of Data Science. 8, 30–62. mla: Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” Foundations of Data Science, vol. 8, AIMS, 2026, pp. 30–62, doi:10.3934/fods.2025003. short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Foundations of Data Science 8 (2026) 30–62. corr_author: '1' date_created: 2025-11-02T23:01:33Z date_published: 2026-03-01T00:00:00Z date_updated: 2026-08-12T06:19:48Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.3934/fods.2025003 ec_funded: 1 external_id: arxiv: - '2212.03128' has_accepted_license: '1' intvolume: ' 8' keyword: - Topological data analysis - Delaunay mosaic - alpha complex - chromatic sets - persistent homology - kernel/image/cokernel persistent homology - radius function - discrete Morse theory - exact sequences language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2212.03128 mathsc: - 62R40 - 55N31 - 68T09 - 57Q70 month: '03' oa: 1 oa_version: Preprint page: 30-62 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Foundations of Data Science publication_identifier: eissn: - 2639-8001 publication_status: published publisher: AIMS quality_controlled: '1' related_material: record: - id: '15091' relation: earlier_version status: public scopus_import: '1' status: public title: Chromatic alpha complexes type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 8 year: '2026' ... --- OA_place: repository OA_type: green _id: '21253' abstract: - lang: eng text: We solve a problem of Dujmović and Wood (2007) by showing that a complete convex geometric graph on n vertices cannot be decomposed into fewer than n - 1 star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs. acknowledgement: A preliminary version of this note has been published in the proceedings of the 31st International Symposium on Graph Drawing and Network Visualization, Palermo, 2023. The authors would like to thank the anonymous referees for their valuable comments. article_number: '102186' article_processing_charge: No article_type: original arxiv: 1 author: - first_name: János full_name: Pach, János last_name: Pach - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian - first_name: Patrick full_name: Schnider, Patrick last_name: Schnider citation: ama: Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests. Computational Geometry. 2025;129. doi:10.1016/j.comgeo.2025.102186 apa: Pach, J., Saghafian, M., & Schnider, P. (2025). Decomposition of geometric graphs into star-forests. Computational Geometry. Elsevier. https://doi.org/10.1016/j.comgeo.2025.102186 chicago: Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric Graphs into Star-Forests.” Computational Geometry. Elsevier, 2025. https://doi.org/10.1016/j.comgeo.2025.102186. ieee: J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs into star-forests,” Computational Geometry, vol. 129. Elsevier, 2025. ista: Pach J, Saghafian M, Schnider P. 2025. Decomposition of geometric graphs into star-forests. Computational Geometry. 129, 102186. mla: Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.” Computational Geometry, vol. 129, 102186, Elsevier, 2025, doi:10.1016/j.comgeo.2025.102186. short: J. Pach, M. Saghafian, P. Schnider, Computational Geometry 129 (2025). corr_author: '1' date_created: 2026-02-16T15:48:42Z date_published: 2025-12-01T00:00:00Z date_updated: 2026-04-16T09:12:36Z day: '01' department: - _id: HeEd doi: 10.1016/j.comgeo.2025.102186 external_id: arxiv: - '2306.13201' intvolume: ' 129' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2306.13201 month: '12' oa: 1 oa_version: Preprint publication: Computational Geometry publication_identifier: issn: - 0925-7721 publication_status: published publisher: Elsevier quality_controlled: '1' related_material: record: - id: '15012' relation: earlier_version status: public status: public title: Decomposition of geometric graphs into star-forests type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 129 year: '2025' ... --- OA_place: repository OA_type: green _id: '18626' abstract: - lang: eng text: "The local angle property of the (order-1) Delaunay triangulations of a generic set in R2\r\n asserts that the sum of two angles opposite a common edge is less than π. This paper extends this property to higher order and uses it to generalize two classic properties from order-1 to order-2: (1) among the complete level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation is the only one that has the local angle property. We also use our method of establishing (2) to give a new short proof of the angle vector optimality for the (order-1) Delaunay triangulation. For order-1, both properties have been instrumental in numerous applications of Delaunay triangulations, and we expect that their generalization will make order-2 Delaunay triangulations more attractive to applications as well." acknowledgement: Work by the first and third authors is partially supported by the European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. Work by the second author is partially supported by the Alexander von Humboldt Foundation. article_number: '110055' article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Alexey full_name: Garber, Alexey last_name: Garber - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Edelsbrunner H, Garber A, Saghafian M. Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. 2025;461. doi:10.1016/j.aim.2024.110055 apa: Edelsbrunner, H., Garber, A., & Saghafian, M. (2025). Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. Elsevier. https://doi.org/10.1016/j.aim.2024.110055 chicago: Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “Order-2 Delaunay Triangulations Optimize Angles.” Advances in Mathematics. Elsevier, 2025. https://doi.org/10.1016/j.aim.2024.110055. ieee: H. Edelsbrunner, A. Garber, and M. Saghafian, “Order-2 Delaunay triangulations optimize angles,” Advances in Mathematics, vol. 461. Elsevier, 2025. ista: Edelsbrunner H, Garber A, Saghafian M. 2025. Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. 461, 110055. mla: Edelsbrunner, Herbert, et al. “Order-2 Delaunay Triangulations Optimize Angles.” Advances in Mathematics, vol. 461, 110055, Elsevier, 2025, doi:10.1016/j.aim.2024.110055. short: H. Edelsbrunner, A. Garber, M. Saghafian, Advances in Mathematics 461 (2025). corr_author: '1' date_created: 2024-12-08T23:01:54Z date_published: 2025-02-01T00:00:00Z date_updated: 2025-04-15T07:16:53Z day: '01' department: - _id: HeEd doi: 10.1016/j.aim.2024.110055 ec_funded: 1 external_id: arxiv: - '2310.18238' isi: - '001370682500001' intvolume: ' 461' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2310.18238 month: '02' oa: 1 oa_version: Preprint project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Advances in Mathematics publication_identifier: eissn: - 1090-2082 issn: - 0001-8708 publication_status: published publisher: Elsevier quality_controlled: '1' scopus_import: '1' status: public title: Order-2 Delaunay triangulations optimize angles type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 461 year: '2025' ... --- OA_type: closed access _id: '19937' abstract: - lang: eng text: Simplets are elementary units within simplicial complexes and are fundamental for analyzing the structure of simplicial complexes. Previous efforts have mainly focused on accurately counting or approximating the number of simplets rather than studying their frequencies. However, analyzing simplet frequencies is more practical for large-scale simplicial complexes. This paper introduces the Simplet Frequency Distribution (SFD) vector, which enables the analysis of simplet frequencies in simplicial complexes. Additionally, we provide a bound on the sample complexity required to approximate the SFD vector using any uniform sampling-based algorithm accurately. We extend the definition of simplet frequency distribution to encompass simplices, allowing for the analysis of simplet frequencies within simplices of simplicial complexes. This paper introduces the Simplet Degree Vector (SDV) and the Simplet Degree Centrality (SDC), facilitating this analysis for each simplex. Furthermore, we present a bound on the sample complexity required for accurately approximating the SDV and SDC for a set of simplices using any uniform sampling-based algorithm. We also introduce algorithms for approximating SFD, geometric SFD, SDV, and SDC. We also validate the theoretical bounds with experiments on random simplicial complexes and demonstrate the practical application through a case study. acknowledgement: "The authors would like to thank the anonymous reviewers for their valuable comments and suggestions, which improved this paper.\r\nWork by the first and fourth authors is partially supported by the European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35." article_number: '122425' article_processing_charge: No article_type: original author: - first_name: Mohammad full_name: Mahini, Mohammad last_name: Mahini - first_name: Hamid full_name: Beigy, Hamid last_name: Beigy - first_name: Salman full_name: Qadami, Salman last_name: Qadami - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: 'Mahini M, Beigy H, Qadami S, Saghafian M. Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. Information Sciences. 2025;719(11). doi:10.1016/j.ins.2025.122425' apa: 'Mahini, M., Beigy, H., Qadami, S., & Saghafian, M. (2025). Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. Information Sciences. Elsevier. https://doi.org/10.1016/j.ins.2025.122425' chicago: 'Mahini, Mohammad, Hamid Beigy, Salman Qadami, and Morteza Saghafian. “Simplet-Based Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.” Information Sciences. Elsevier, 2025. https://doi.org/10.1016/j.ins.2025.122425.' ieee: 'M. Mahini, H. Beigy, S. Qadami, and M. Saghafian, “Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality,” Information Sciences, vol. 719, no. 11. Elsevier, 2025.' ista: 'Mahini M, Beigy H, Qadami S, Saghafian M. 2025. Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. Information Sciences. 719(11), 122425.' mla: 'Mahini, Mohammad, et al. “Simplet-Based Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.” Information Sciences, vol. 719, no. 11, 122425, Elsevier, 2025, doi:10.1016/j.ins.2025.122425.' short: M. Mahini, H. Beigy, S. Qadami, M. Saghafian, Information Sciences 719 (2025). corr_author: '1' date_created: 2025-06-30T08:48:48Z date_published: 2025-11-01T00:00:00Z date_updated: 2025-12-30T09:05:32Z day: '01' department: - _id: HeEd doi: 10.1016/j.ins.2025.122425 ec_funded: 1 external_id: isi: - '001516170500002' intvolume: ' 719' isi: 1 issue: '11' language: - iso: eng month: '11' oa_version: None project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Information Sciences publication_identifier: issn: - 0020-0255 publication_status: published publisher: Elsevier quality_controlled: '1' scopus_import: '1' status: public title: 'Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality' type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 719 year: '2025' ... --- OA_place: publisher OA_type: gold _id: '20005' abstract: - lang: eng text: "We generalize a classical result by Boris Delaunay that introduced Delaunay triangulations. In particular, we prove that for a locally finite and coarsely dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose exactly k points of A. We extend this result to the cases in which the points are weighted, and when A contains only finitely many points in ℝ^d or in \U0001D54A^d. Furthermore, we use the result to give a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers." acknowledgement: "Herbert Edelsbrunner: partially supported by the Wittgenstein Prize, Austrian Science\r\nFund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109,\r\nAustrian Science Fund (FWF), grant no. I 02979-N35.\r\nAlexey Garber: partially supported by the Simons Foundation.\r\nMorteza Saghafian: partially supported by the Wittgenstein Prize, Austrian Science Fund (FWF),\r\ngrant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science\r\nFund (FWF), grant no. I 02979-N35" alternative_title: - LIPIcs article_number: '43' article_processing_charge: Yes arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Alexey full_name: Garber, Alexey last_name: Garber - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: 'Edelsbrunner H, Garber A, Saghafian M. On spheres with k points inside. In: 41st International Symposium on Computational Geometry. Vol 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:10.4230/LIPIcs.SoCG.2025.43' apa: 'Edelsbrunner, H., Garber, A., & Saghafian, M. (2025). On spheres with k points inside. In 41st International Symposium on Computational Geometry (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2025.43' chicago: Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “On Spheres with k Points Inside.” In 41st International Symposium on Computational Geometry, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. https://doi.org/10.4230/LIPIcs.SoCG.2025.43. ieee: H. Edelsbrunner, A. Garber, and M. Saghafian, “On spheres with k points inside,” in 41st International Symposium on Computational Geometry, Kanazawa, Japan, 2025, vol. 332. ista: 'Edelsbrunner H, Garber A, Saghafian M. 2025. On spheres with k points inside. 41st International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 43.' mla: Edelsbrunner, Herbert, et al. “On Spheres with k Points Inside.” 41st International Symposium on Computational Geometry, vol. 332, 43, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:10.4230/LIPIcs.SoCG.2025.43. short: H. Edelsbrunner, A. Garber, M. Saghafian, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. conference: end_date: 2025-06-27 location: Kanazawa, Japan name: 'SoCG: Symposium on Computational Geometry' start_date: 2025-06-23 corr_author: '1' date_created: 2025-07-13T22:01:22Z date_published: 2025-06-20T00:00:00Z date_updated: 2025-07-14T07:26:14Z day: '20' ddc: - '510' department: - _id: HeEd doi: 10.4230/LIPIcs.SoCG.2025.43 external_id: arxiv: - '2410.21204' file: - access_level: open_access checksum: b5313ed8575ea87913c71a6e3c7513c8 content_type: application/pdf creator: dernst date_created: 2025-07-14T07:24:22Z date_updated: 2025-07-14T07:24:22Z file_id: '20016' file_name: 2025_LIPIcs.SoCG_Edelsbrunner.pdf file_size: 661893 relation: main_file success: 1 file_date_updated: 2025-07-14T07:24:22Z has_accepted_license: '1' intvolume: ' 332' language: - iso: eng month: '06' oa: 1 oa_version: Published Version project: - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: 41st International Symposium on Computational Geometry publication_identifier: eissn: - 1868-8969 isbn: - '9783959773706' publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' scopus_import: '1' status: public title: On spheres with k points inside 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: 332 year: '2025' ... --- _id: '14345' abstract: - lang: eng text: For a locally finite set in R2, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2 is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles (Math. Biosci. 6, 85–127 (1970)). acknowledgement: Work by all authors but A. Garber is supported by the European Research Council (ERC), Grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), Grant No. I 02979-N35. Work by A. Garber is partially supported by the Alexander von Humboldt Foundation. article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Alexey full_name: Garber, Alexey last_name: Garber - first_name: Mohadese full_name: Ghafari, Mohadese last_name: Ghafari - first_name: Teresa full_name: Heiss, Teresa id: 4879BB4E-F248-11E8-B48F-1D18A9856A87 last_name: Heiss orcid: 0000-0002-1780-2689 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. On angles in higher order Brillouin tessellations and related tilings in the plane. Discrete and Computational Geometry. 2024;72:29-48. doi:10.1007/s00454-023-00566-1 apa: Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., & Saghafian, M. (2024). On angles in higher order Brillouin tessellations and related tilings in the plane. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-023-00566-1 chicago: Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” Discrete and Computational Geometry. Springer Nature, 2024. https://doi.org/10.1007/s00454-023-00566-1. ieee: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “On angles in higher order Brillouin tessellations and related tilings in the plane,” Discrete and Computational Geometry, vol. 72. Springer Nature, pp. 29–48, 2024. ista: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2024. On angles in higher order Brillouin tessellations and related tilings in the plane. Discrete and Computational Geometry. 72, 29–48. mla: Edelsbrunner, Herbert, et al. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” Discrete and Computational Geometry, vol. 72, Springer Nature, 2024, pp. 29–48, doi:10.1007/s00454-023-00566-1. short: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete and Computational Geometry 72 (2024) 29–48. corr_author: '1' date_created: 2023-09-17T22:01:10Z date_published: 2024-07-01T00:00:00Z date_updated: 2025-04-23T08:41:59Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.1007/s00454-023-00566-1 ec_funded: 1 external_id: arxiv: - '2204.01076' isi: - '001060727600004' pmid: - '39610762' file: - access_level: open_access checksum: b207b4e00f904e8ea8a30e24f0251f79 content_type: application/pdf creator: dernst date_created: 2024-07-22T09:43:19Z date_updated: 2024-07-22T09:43:19Z file_id: '17301' file_name: 2024_DiscreteComputGeom_Edelsbrunner.pdf file_size: 892019 relation: main_file success: 1 file_date_updated: 2024-07-22T09:43:19Z has_accepted_license: '1' intvolume: ' 72' isi: 1 language: - iso: eng month: '07' oa: 1 oa_version: Published Version page: 29-48 pmid: 1 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Discrete and Computational Geometry publication_identifier: eissn: - 1432-0444 issn: - 0179-5376 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: On angles in higher order Brillouin tessellations and related tilings in the plane 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: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 72 year: '2024' ... --- _id: '15012' abstract: - lang: eng text: We solve a problem of Dujmović and Wood (2007) by showing that a complete convex geometric graph on n vertices cannot be decomposed into fewer than n-1 star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs. acknowledgement: János Pach’s Research partially supported by European Research Council (ERC), grant “GeoScape” No. 882971 and by the Hungarian Science Foundation (NKFIH), grant K-131529. Work by Morteza Saghafian is partially supported by the European Research Council (ERC), grant No. 788183, and by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31. alternative_title: - LNCS article_processing_charge: No arxiv: 1 author: - first_name: János full_name: Pach, János id: E62E3130-B088-11EA-B919-BF823C25FEA4 last_name: Pach - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian - first_name: Patrick full_name: Schnider, Patrick last_name: Schnider citation: ama: 'Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests. In: 31st International Symposium on Graph Drawing and Network Visualization. Vol 14465. Springer Nature; 2024:339-346. doi:10.1007/978-3-031-49272-3_23' apa: 'Pach, J., Saghafian, M., & Schnider, P. (2024). Decomposition of geometric graphs into star-forests. In 31st International Symposium on Graph Drawing and Network Visualization (Vol. 14465, pp. 339–346). Isola delle Femmine, Palermo, Italy: Springer Nature. https://doi.org/10.1007/978-3-031-49272-3_23' chicago: Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric Graphs into Star-Forests.” In 31st International Symposium on Graph Drawing and Network Visualization, 14465:339–46. Springer Nature, 2024. https://doi.org/10.1007/978-3-031-49272-3_23. ieee: J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs into star-forests,” in 31st International Symposium on Graph Drawing and Network Visualization, Isola delle Femmine, Palermo, Italy, 2024, vol. 14465, pp. 339–346. ista: 'Pach J, Saghafian M, Schnider P. 2024. Decomposition of geometric graphs into star-forests. 31st International Symposium on Graph Drawing and Network Visualization. GD: Graph Drawing and Network Visualization, LNCS, vol. 14465, 339–346.' mla: Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.” 31st International Symposium on Graph Drawing and Network Visualization, vol. 14465, Springer Nature, 2024, pp. 339–46, doi:10.1007/978-3-031-49272-3_23. short: J. Pach, M. Saghafian, P. Schnider, in:, 31st International Symposium on Graph Drawing and Network Visualization, Springer Nature, 2024, pp. 339–346. conference: end_date: 2023-09-22 location: Isola delle Femmine, Palermo, Italy name: 'GD: Graph Drawing and Network Visualization' start_date: 2023-09-20 date_created: 2024-02-18T23:01:03Z date_published: 2024-01-01T00:00:00Z date_updated: 2026-04-16T09:12:37Z day: '01' department: - _id: HeEd doi: 10.1007/978-3-031-49272-3_23 ec_funded: 1 external_id: arxiv: - '2306.13201' isi: - '001207939600023' intvolume: ' 14465' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2306.13201 month: '01' oa: 1 oa_version: Preprint page: 339-346 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science publication: 31st International Symposium on Graph Drawing and Network Visualization publication_identifier: eisbn: - '9783031492723' eissn: - 1611-3349 isbn: - '9783031492716' issn: - 0302-9743 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '21253' relation: later_version status: public scopus_import: '1' status: public title: Decomposition of geometric graphs into star-forests type: conference user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd volume: 14465 year: '2024' ... --- OA_place: publisher OA_type: hybrid _id: '15380' abstract: - lang: eng text: The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements. acknowledgement: "The authors thank Uli Wagner and Emo Welzl for comments on an earlier version of this paper, and for pointing out related work in the prior literature.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35." article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Sebastiano full_name: Cultrera Di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera Di Montesano orcid: 0000-0001-6249-0832 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Journal of Applied and Computational Topology. 2024;8:557-578. doi:10.1007/s41468-024-00173-w' apa: 'Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (2024). Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-024-00173-w' chicago: 'Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” Journal of Applied and Computational Topology. Springer Nature, 2024. https://doi.org/10.1007/s41468-024-00173-w.' ieee: 'R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth in arrangements: Dehn–Sommerville–Euler relations with applications,” Journal of Applied and Computational Topology, vol. 8. Springer Nature, pp. 557–578, 2024.' ista: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2024. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Journal of Applied and Computational Topology. 8, 557–578.' mla: 'Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” Journal of Applied and Computational Topology, vol. 8, Springer Nature, 2024, pp. 557–78, doi:10.1007/s41468-024-00173-w.' short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology 8 (2024) 557–578. corr_author: '1' date_created: 2024-05-12T22:01:03Z date_published: 2024-09-01T00:00:00Z date_updated: 2025-05-14T09:27:57Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.1007/s41468-024-00173-w ec_funded: 1 external_id: pmid: - '39308789' file: - access_level: open_access checksum: 0ee15c1493a6413cf356ab2f32c81a9e content_type: application/pdf creator: dernst date_created: 2025-04-23T08:01:36Z date_updated: 2025-04-23T08:01:36Z file_id: '19612' file_name: 2024_JourApplCompTopo_BiswasRa.pdf file_size: 522831 relation: main_file success: 1 file_date_updated: 2025-04-23T08:01:36Z has_accepted_license: '1' intvolume: ' 8' language: - iso: eng month: '09' oa: 1 oa_version: Published Version page: 557-578 pmid: 1 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Journal of Applied and Computational Topology publication_identifier: eissn: - 2367-1734 issn: - 2367-1726 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '11658' relation: earlier_version status: public scopus_import: '1' status: public title: 'Depth in arrangements: Dehn–Sommerville–Euler relations with applications' 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: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 8 year: '2024' ... --- _id: '17145' abstract: - lang: eng text: Grid peeling is the process of repeatedly removing the convex hull vertices of the grid points that lie inside a given convex curve. It has been conjectured that, for a more and more refined grid, grid peeling converges to a continuous process, the affine curve-shortening flow, which deforms the curve based on the curvature. We prove this conjecture for one class of curves, parabolas with a vertical axis, and we determine the value of the constant factor in the formula that relates the two processes. acknowledgement: Part of this work was done while G.R. enjoyed the hospitality of the Institute of Science and Technology Austria (ISTA) as a visiting professor during his sabbatical in the winter semester 2022/23. alternative_title: - LIPIcs article_number: '76' article_processing_charge: No arxiv: 1 author: - first_name: Günter full_name: Rote, Günter last_name: Rote - first_name: Moritz full_name: Rüber, Moritz last_name: Rüber - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: 'Rote G, Rüber M, Saghafian M. Grid peeling of parabolas. In: 40th International Symposium on Computational Geometry. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.76' apa: 'Rote, G., Rüber, M., & Saghafian, M. (2024). Grid peeling of parabolas. In 40th International Symposium on Computational Geometry (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.76' chicago: Rote, Günter, Moritz Rüber, and Morteza Saghafian. “Grid Peeling of Parabolas.” In 40th International Symposium on Computational Geometry, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.76. ieee: G. Rote, M. Rüber, and M. Saghafian, “Grid peeling of parabolas,” in 40th International Symposium on Computational Geometry, Athens, Greece, 2024, vol. 293. ista: 'Rote G, Rüber M, Saghafian M. 2024. Grid peeling of parabolas. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 76.' mla: Rote, Günter, et al. “Grid Peeling of Parabolas.” 40th International Symposium on Computational Geometry, vol. 293, 76, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.76. short: G. Rote, M. Rüber, M. Saghafian, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. conference: end_date: 2024-06-14 location: Athens, Greece name: 'SoCG: Symposium on Computational Geometry' start_date: 2024-06-11 date_created: 2024-06-16T22:01:06Z date_published: 2024-06-01T00:00:00Z date_updated: 2024-06-17T08:41:56Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.4230/LIPIcs.SoCG.2024.76 external_id: arxiv: - '2402.15787' file: - access_level: open_access checksum: fbad1de06383a6b7e8a1cb3e8c7205ce content_type: application/pdf creator: dernst date_created: 2024-06-17T08:40:04Z date_updated: 2024-06-17T08:40:04Z file_id: '17151' file_name: 2024_LIPICS_Rote.pdf file_size: 1430896 relation: main_file success: 1 file_date_updated: 2024-06-17T08:40:04Z has_accepted_license: '1' intvolume: ' 293' language: - iso: eng month: '06' oa: 1 oa_version: Published Version publication: 40th International Symposium on Computational Geometry publication_identifier: isbn: - '9783959773164' issn: - 1868-8969 publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' scopus_import: '1' status: public title: Grid peeling of parabolas 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: 293 year: '2024' ... --- OA_place: publisher OA_type: gold _id: '18556' abstract: - lang: eng text: Given a finite set, A ⊆ ℝ², and a subset, B ⊆ A, the MST-ratio is the combined length of the minimum spanning trees of B and A⧵B divided by the length of the minimum spanning tree of A. The question of the supremum, over all sets A, of the maximum, over all subsets B, is related to the Steiner ratio, and we prove this sup-max is between 2.154 and 2.427. Restricting ourselves to 2-dimensional lattices, we prove that the sup-max is 2, while the inf-max is 1.25. By some margin the most difficult of these results is the upper bound for the inf-max, which we prove by showing that the hexagonal lattice cannot have MST-ratio larger than 1.25. acknowledgement: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, "Discretization in Geometry and Dynamics", Austrian Science Fund (FWF), grant no. I 02979-N35. alternative_title: - LIPIcs article_number: '3' article_processing_charge: Yes arxiv: 1 author: - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: 'Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. The Euclidean MST-ratio for bi-colored lattices. In: 32nd International Symposium on Graph Drawing and Network Visualization. Vol 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.GD.2024.3' apa: 'Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (2024). The Euclidean MST-ratio for bi-colored lattices. In 32nd International Symposium on Graph Drawing and Network Visualization (Vol. 320). Vienna, Austria: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.GD.2024.3' chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “The Euclidean MST-Ratio for Bi-Colored Lattices.” In 32nd International Symposium on Graph Drawing and Network Visualization, Vol. 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.GD.2024.3. ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “The Euclidean MST-ratio for bi-colored lattices,” in 32nd International Symposium on Graph Drawing and Network Visualization, Vienna, Austria, 2024, vol. 320. ista: 'Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2024. The Euclidean MST-ratio for bi-colored lattices. 32nd International Symposium on Graph Drawing and Network Visualization. GD: Graph Drawing and Network Visualization, LIPIcs, vol. 320, 3.' mla: Cultrera di Montesano, Sebastiano, et al. “The Euclidean MST-Ratio for Bi-Colored Lattices.” 32nd International Symposium on Graph Drawing and Network Visualization, vol. 320, 3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.GD.2024.3. short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, in:, 32nd International Symposium on Graph Drawing and Network Visualization, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. conference: end_date: 2024-09-20 location: Vienna, Austria name: 'GD: Graph Drawing and Network Visualization' start_date: 2024-09-18 corr_author: '1' date_created: 2024-11-17T23:01:47Z date_published: 2024-10-28T00:00:00Z date_updated: 2025-12-02T13:50:50Z day: '28' ddc: - '510' department: - _id: HeEd doi: 10.4230/LIPIcs.GD.2024.3 ec_funded: 1 external_id: arxiv: - '2403.10204' isi: - '001540278400001' file: - access_level: open_access checksum: 5f9b35e115c3d375e99be78da9054cb4 content_type: application/pdf creator: dernst date_created: 2024-11-18T07:49:25Z date_updated: 2024-11-18T07:49:25Z file_id: '18560' file_name: 2024_LIPIcs_CultreradiMontesano.pdf file_size: 908541 relation: main_file success: 1 file_date_updated: 2024-11-18T07:49:25Z has_accepted_license: '1' intvolume: ' 320' isi: 1 language: - iso: eng month: '10' oa: 1 oa_version: Published Version project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: 32nd International Symposium on Graph Drawing and Network Visualization publication_identifier: isbn: - '9783959773430' issn: - 1868-8969 publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' scopus_import: '1' status: public title: The Euclidean MST-ratio for bi-colored lattices 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: 320 year: '2024' ... --- OA_place: repository OA_type: green _id: '18999' abstract: - lang: eng text: Exploring the shape of point configurations has been a key driver in the evolution of TDA (short for topological data analysis) since its infancy. This survey illustrates the recent efforts to broaden these ideas to model spatial interactions among multiple configurations, each distinguished by a color. It describes advances in this area and prepares the ground for further exploration by mentioning unresolved questions and promising research avenues while focusing on the overlap with discrete geometry. article_number: '2406.04102' article_processing_charge: No arxiv: 1 author: - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic topological data analysis. arXiv. doi:10.48550/ARXIV.2406.04102 apa: Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (n.d.). Chromatic topological data analysis. arXiv. https://doi.org/10.48550/ARXIV.2406.04102 chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Topological Data Analysis.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2406.04102. ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic topological data analysis,” arXiv. . ista: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic topological data analysis. arXiv, 2406.04102. mla: Cultrera di Montesano, Sebastiano, et al. “Chromatic Topological Data Analysis.” ArXiv, 2406.04102, doi:10.48550/ARXIV.2406.04102. short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.). corr_author: '1' date_created: 2025-02-04T16:21:21Z date_published: 2024-06-06T00:00:00Z date_updated: 2025-02-10T08:14:27Z day: '06' ddc: - '510' department: - _id: GradSch - _id: HeEd doi: 10.48550/ARXIV.2406.04102 external_id: arxiv: - '2406.04102' has_accepted_license: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2406.04102 month: '06' oa: 1 oa_version: Preprint publication: arXiv publication_status: submitted status: public title: Chromatic topological data analysis 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: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2024' ... --- OA_place: repository _id: '15091' abstract: - lang: eng text: "Motivated by applications in the medical sciences, we study finite chromatic\r\nsets in Euclidean space from a topological perspective. Based on the persistent\r\nhomology for images, kernels and cokernels, we design provably stable\r\nhomological quantifiers that describe the geometric micro- and macro-structure\r\nof how the color classes mingle. These can be efficiently computed using\r\nchromatic variants of Delaunay and alpha complexes, and code that does these\r\ncomputations is provided." article_number: '2212.03128' article_processing_charge: No arxiv: 1 author: - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. arXiv. doi:10.48550/arXiv.2212.03128 apa: Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (n.d.). Chromatic alpha complexes. arXiv. https://doi.org/10.48550/arXiv.2212.03128 chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Alpha Complexes.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2212.03128. ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic alpha complexes,” arXiv. . ista: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. arXiv, 2212.03128. mla: Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” ArXiv, 2212.03128, doi:10.48550/arXiv.2212.03128. short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.). corr_author: '1' date_created: 2024-03-08T10:13:59Z date_published: 2024-02-07T00:00:00Z date_updated: 2026-08-12T06:19:48Z day: '07' department: - _id: HeEd doi: 10.48550/arXiv.2212.03128 external_id: arxiv: - '2212.03128' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2212.03128 month: '02' oa: 1 oa_version: Preprint publication: arXiv publication_status: draft related_material: record: - id: '18979' relation: dissertation_contains status: public - id: '15094' relation: dissertation_contains status: public - id: '20585' relation: later_version status: public status: public title: Chromatic alpha complexes 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: preprint user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 year: '2024' ... --- OA_place: publisher OA_type: hybrid _id: '13182' abstract: - lang: eng text: "We characterize critical points of 1-dimensional maps paired in persistent homology\r\ngeometrically and this way get elementary proofs of theorems about the symmetry\r\nof persistence diagrams and the variation of such maps. In particular, we identify\r\nbranching points and endpoints of networks as the sole source of asymmetry and\r\nrelate the cycle basis in persistent homology with a version of the stable marriage\r\nproblem. Our analysis provides the foundations of fast algorithms for maintaining a\r\ncollection of sorted lists together with its persistence diagram." acknowledgement: Open access funding provided by Austrian Science Fund (FWF). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35. The authors of this paper thank anonymous reviewers for their constructive criticism and Monika Henzinger for detailed comments on an earlier version of this paper. article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Sebastiano full_name: Cultrera Di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera Di Montesano orcid: 0000-0001-6249-0832 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Geometric characterization of the persistence of 1D maps. Journal of Applied and Computational Topology. 2024;8:1101-1119. doi:10.1007/s41468-023-00126-9 apa: Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (2024). Geometric characterization of the persistence of 1D maps. Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-023-00126-9 chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Geometric Characterization of the Persistence of 1D Maps.” Journal of Applied and Computational Topology. Springer Nature, 2024. https://doi.org/10.1007/s41468-023-00126-9. ieee: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Geometric characterization of the persistence of 1D maps,” Journal of Applied and Computational Topology, vol. 8. Springer Nature, pp. 1101–1119, 2024. ista: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2024. Geometric characterization of the persistence of 1D maps. Journal of Applied and Computational Topology. 8, 1101–1119. mla: Biswas, Ranita, et al. “Geometric Characterization of the Persistence of 1D Maps.” Journal of Applied and Computational Topology, vol. 8, Springer Nature, 2024, pp. 1101–19, doi:10.1007/s41468-023-00126-9. short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology 8 (2024) 1101–1119. corr_author: '1' date_created: 2023-07-02T22:00:44Z date_published: 2024-10-01T00:00:00Z date_updated: 2026-08-13T09:41:57Z day: '01' ddc: - '000' department: - _id: HeEd doi: 10.1007/s41468-023-00126-9 ec_funded: 1 external_id: pmid: - '39678706' file: - access_level: open_access checksum: d493df5088c222b88d9ca46b623ad0ee content_type: application/pdf creator: dernst date_created: 2025-01-09T07:39:41Z date_updated: 2025-01-09T07:39:41Z file_id: '18783' file_name: 2024_JourApplCompTopo_Biswas.pdf file_size: 476896 relation: main_file success: 1 file_date_updated: 2025-01-09T07:39:41Z has_accepted_license: '1' intvolume: ' 8' language: - iso: eng month: '10' oa: 1 oa_version: Published Version page: 1101-1119 pmid: 1 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316 grant_number: I4887 name: Persistent Homology, Algorithms and Stochastic Geometry - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science publication: Journal of Applied and Computational Topology publication_identifier: eissn: - 2367-1734 issn: - 2367-1726 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '15094' relation: dissertation_contains status: public - id: '11660' relation: earlier_version status: public scopus_import: '1' status: public title: Geometric characterization of the persistence of 1D maps 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: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 8 year: '2024' ... --- _id: '11658' abstract: - lang: eng text: The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements. acknowledgement: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35. article_processing_charge: No author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Leibniz International Proceedings on Mathematics.' apa: 'Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (n.d.). Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Leibniz International Proceedings on Mathematics. Schloss Dagstuhl - Leibniz-Zentrum für Informatik.' chicago: 'Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” Leibniz International Proceedings on Mathematics. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, n.d.' ieee: 'R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth in arrangements: Dehn–Sommerville–Euler relations with applications,” Leibniz International Proceedings on Mathematics. Schloss Dagstuhl - Leibniz-Zentrum für Informatik.' ista: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Leibniz International Proceedings on Mathematics.' mla: 'Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” Leibniz International Proceedings on Mathematics, Schloss Dagstuhl - Leibniz-Zentrum für Informatik.' short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Leibniz International Proceedings on Mathematics (n.d.). corr_author: '1' date_created: 2022-07-27T09:27:34Z date_published: 2022-07-27T00:00:00Z date_updated: 2026-04-07T12:58:48Z day: '27' ddc: - '510' department: - _id: GradSch - _id: HeEd ec_funded: 1 file: - access_level: open_access checksum: b2f511e8b1cae5f1892b0cdec341acac content_type: application/pdf creator: scultrer date_created: 2022-07-27T09:25:53Z date_updated: 2022-07-27T09:25:53Z file_id: '11659' file_name: D-S-E.pdf file_size: 639266 relation: main_file file_date_updated: 2022-07-27T09:25:53Z has_accepted_license: '1' language: - iso: eng month: '07' oa: 1 oa_version: Submitted Version project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Leibniz International Proceedings on Mathematics publication_status: draft publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' related_material: record: - id: '15380' relation: later_version status: public - id: '15094' relation: dissertation_contains status: public status: public title: 'Depth in arrangements: Dehn–Sommerville–Euler relations with applications' 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: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2022' ... --- OA_place: repository _id: '15090' abstract: - lang: eng text: Given a locally finite set A⊆Rd and a coloring χ:A→{0,1,…,s}, we introduce the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in Rs+d that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with n=#A, and the coloring is random, then the chromatic Delaunay mosaic has O(n⌈d/2⌉) cells in expectation. In contrast, for Delone sets and Poisson point processes in Rd, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in R2 all colorings of a dense set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications. article_number: '2212.03121' article_processing_charge: No arxiv: 1 author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. arXiv. apa: Biswas, R., Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (n.d.). On the size of chromatic Delaunay mosaics. arXiv. chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “On the Size of Chromatic Delaunay Mosaics.” ArXiv, n.d. ieee: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “On the size of chromatic Delaunay mosaics,” arXiv. . ista: Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. arXiv, 2212.03121. mla: Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” ArXiv, 2212.03121. short: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.). corr_author: '1' date_created: 2024-03-08T09:54:20Z date_published: 2022-12-06T00:00:00Z date_updated: 2026-04-07T12:58:47Z day: '06' department: - _id: HeEd ec_funded: 1 external_id: arxiv: - '2212.03121' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2212.03121 month: '12' oa: 1 oa_version: Preprint project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316 grant_number: I4887 name: Persistent Homology, Algorithms and Stochastic Geometry - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science publication: arXiv publication_status: draft related_material: record: - id: '20456' relation: later_version status: public - id: '15094' relation: dissertation_contains status: public status: public title: On the size of chromatic Delaunay mosaics 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: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2022' ...