[{"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.","oa":1,"doi":"10.1007/s00454-025-00778-7","author":[{"id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","first_name":"Ranita","orcid":"0000-0002-5372-7890","full_name":"Biswas, Ranita","last_name":"Biswas"},{"orcid":"0000-0001-6249-0832","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano","last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano"},{"last_name":"Draganov","full_name":"Draganov, Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","first_name":"Ondrej","orcid":"0000-0003-0464-3823"},{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"volume":75,"article_type":"original","project":[{"name":"Alpha Shape Theory Extended","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"month":"01","department":[{"_id":"HeEd"}],"tmp":{"short":"CC BY (4.0)","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)"},"corr_author":"1","article_processing_charge":"Yes (via OA deal)","publisher":"Springer Nature","related_material":{"record":[{"relation":"earlier_version","status":"public","id":"15090"}]},"publication_status":"published","file":[{"content_type":"application/pdf","success":1,"file_size":570922,"date_created":"2026-01-05T13:21:20Z","access_level":"open_access","date_updated":"2026-01-05T13:21:20Z","file_id":"20952","creator":"dernst","relation":"main_file","checksum":"0addb5c1b78142f9fb453bfa04695400","file_name":"2026_DiscreteCompGeom_Biswas.pdf"}],"arxiv":1,"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"title":"On the size of chromatic Delaunay mosaics","date_created":"2025-10-12T22:01:26Z","oa_version":"Published Version","OA_type":"hybrid","file_date_updated":"2026-01-05T13:21:20Z","publication":"Discrete and Computational Geometry","date_updated":"2026-01-05T13:21:56Z","isi":1,"PlanS_conform":"1","page":"24-47","status":"public","ec_funded":1,"scopus_import":"1","abstract":[{"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.","lang":"eng"}],"_id":"20456","language":[{"iso":"eng"}],"external_id":{"arxiv":["2212.03121"],"isi":["001584166900001"]},"type":"journal_article","quality_controlled":"1","has_accepted_license":"1","day":"01","citation":{"apa":"Biswas, R., Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2026). On the size of chromatic Delaunay mosaics. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-025-00778-7\">https://doi.org/10.1007/s00454-025-00778-7</a>","ieee":"R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “On the size of chromatic Delaunay mosaics,” <i>Discrete and Computational Geometry</i>, vol. 75. Springer Nature, pp. 24–47, 2026.","short":"R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Discrete and Computational Geometry 75 (2026) 24–47.","mla":"Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” <i>Discrete and Computational Geometry</i>, vol. 75, Springer Nature, 2026, pp. 24–47, doi:<a href=\"https://doi.org/10.1007/s00454-025-00778-7\">10.1007/s00454-025-00778-7</a>.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “On the Size of Chromatic Delaunay Mosaics.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/s00454-025-00778-7\">https://doi.org/10.1007/s00454-025-00778-7</a>.","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.","ama":"Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. <i>Discrete and Computational Geometry</i>. 2026;75:24-47. doi:<a href=\"https://doi.org/10.1007/s00454-025-00778-7\">10.1007/s00454-025-00778-7</a>"},"intvolume":"        75","year":"2026","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ddc":["510"],"date_published":"2026-01-01T00:00:00Z","OA_place":"publisher"},{"OA_place":"repository","date_published":"2026-02-13T00:00:00Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2511.22526","open_access":"1"}],"intvolume":"     16448","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2026","citation":{"mla":"Antić, Todor, et al. “Edge-Constrained Hamiltonian Paths on a Point Set.” <i>51st International Conference on Current Trends in Theory and Practice of Computer Science</i>, vol. 16448, Springer Nature, 2026, pp. 532–46, doi:<a href=\"https://doi.org/10.1007/978-3-032-17801-5_39\">10.1007/978-3-032-17801-5_39</a>.","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.","ieee":"T. Antić <i>et al.</i>, “Edge-constrained Hamiltonian paths on a point set,” in <i>51st International Conference on Current Trends in Theory and Practice of Computer Science</i>, Krakow, Poland, 2026, vol. 16448, pp. 532–546.","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 <i>51st International Conference on Current Trends in Theory and Practice of Computer Science</i> (Vol. 16448, pp. 532–546). Krakow, Poland: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-032-17801-5_39\">https://doi.org/10.1007/978-3-032-17801-5_39</a>","ama":"Antić T, Džuklevski A, Fiala J, et al. Edge-constrained Hamiltonian paths on a point set. In: <i>51st International Conference on Current Trends in Theory and Practice of Computer Science</i>. Vol 16448. Springer Nature; 2026:532-546. doi:<a href=\"https://doi.org/10.1007/978-3-032-17801-5_39\">10.1007/978-3-032-17801-5_39</a>","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.","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 <i>51st International Conference on Current Trends in Theory and Practice of Computer Science</i>, 16448:532–46. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/978-3-032-17801-5_39\">https://doi.org/10.1007/978-3-032-17801-5_39</a>."},"day":"13","external_id":{"arxiv":["2511.22526"]},"_id":"21374","language":[{"iso":"eng"}],"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."}],"quality_controlled":"1","type":"conference","scopus_import":"1","status":"public","page":"532-546","date_updated":"2026-03-02T08:49:20Z","publication":"51st International Conference on Current Trends in Theory and Practice of Computer Science","date_created":"2026-03-01T23:01:40Z","title":"Edge-constrained Hamiltonian paths on a point set","alternative_title":["LNCS"],"OA_type":"green","oa_version":"Preprint","publication_identifier":{"isbn":["9783032178008"],"issn":["0302-9743"],"eissn":["1611-3349"]},"arxiv":1,"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"},"publication_status":"published","article_processing_charge":"No","publisher":"Springer Nature","author":[{"first_name":"Todor","last_name":"Antić","full_name":"Antić, Todor"},{"first_name":"Aleksa","full_name":"Džuklevski, Aleksa","last_name":"Džuklevski"},{"full_name":"Fiala, Jiří","last_name":"Fiala","first_name":"Jiří"},{"first_name":"Jan","full_name":"Kratochvíl, Jan","last_name":"Kratochvíl"},{"first_name":"Giuseppe","last_name":"Liotta","full_name":"Liotta, Giuseppe"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"},{"last_name":"Saumell","full_name":"Saumell, Maria","first_name":"Maria"},{"last_name":"Zink","full_name":"Zink, Johannes","first_name":"Johannes"}],"volume":16448,"month":"02","department":[{"_id":"HeEd"}],"doi":"10.1007/978-3-032-17801-5_39","oa":1,"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."},{"status":"public","page":"386-401","ec_funded":1,"scopus_import":"1","date_updated":"2026-03-09T10:25:41Z","publication":"20th International Conference and Workshops on Algorithms and Computation","intvolume":"     16444","year":"2026","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2026-02-14T00:00:00Z","OA_place":"repository","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2409.11079","open_access":"1"}],"language":[{"iso":"eng"}],"_id":"21410","abstract":[{"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.","lang":"eng"}],"external_id":{"arxiv":["2409.11079"]},"type":"conference","quality_controlled":"1","day":"14","citation":{"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.","ama":"Jabal Ameli A, Motiei F, Saghafian M. On the MST-ratio: Theoretical bounds and complexity of finding the maximum. In: <i>20th International Conference and Workshops on Algorithms and Computation</i>. Vol 16444. Springer Nature; 2026:386-401. doi:<a href=\"https://doi.org/10.1007/978-981-95-7127-7_26\">10.1007/978-981-95-7127-7_26</a>","chicago":"Jabal Ameli, Afrouz, Faezeh Motiei, and Morteza Saghafian. “On the MST-Ratio: Theoretical Bounds and Complexity of Finding the Maximum.” In <i>20th International Conference and Workshops on Algorithms and Computation</i>, 16444:386–401. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/978-981-95-7127-7_26\">https://doi.org/10.1007/978-981-95-7127-7_26</a>.","short":"A. Jabal Ameli, F. Motiei, M. Saghafian, in:, 20th International Conference and Workshops on Algorithms and Computation, Springer Nature, 2026, pp. 386–401.","ieee":"A. Jabal Ameli, F. Motiei, and M. Saghafian, “On the MST-ratio: Theoretical bounds and complexity of finding the maximum,” in <i>20th International Conference and Workshops on Algorithms and Computation</i>, Perugia, Italy, 2026, vol. 16444, pp. 386–401.","mla":"Jabal Ameli, Afrouz, et al. “On the MST-Ratio: Theoretical Bounds and Complexity of Finding the Maximum.” <i>20th International Conference and Workshops on Algorithms and Computation</i>, vol. 16444, Springer Nature, 2026, pp. 386–401, doi:<a href=\"https://doi.org/10.1007/978-981-95-7127-7_26\">10.1007/978-981-95-7127-7_26</a>.","apa":"Jabal Ameli, A., Motiei, F., &#38; Saghafian, M. (2026). On the MST-ratio: Theoretical bounds and complexity of finding the maximum. In <i>20th International Conference and Workshops on Algorithms and Computation</i> (Vol. 16444, pp. 386–401). Perugia, Italy: Springer Nature. <a href=\"https://doi.org/10.1007/978-981-95-7127-7_26\">https://doi.org/10.1007/978-981-95-7127-7_26</a>"},"article_processing_charge":"No","publisher":"Springer Nature","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.","doi":"10.1007/978-981-95-7127-7_26","oa":1,"volume":16444,"author":[{"first_name":"Afrouz","last_name":"Jabal Ameli","full_name":"Jabal Ameli, Afrouz"},{"last_name":"Motiei","full_name":"Motiei, Faezeh","first_name":"Faezeh"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science"}],"month":"02","department":[{"_id":"HeEd"}],"arxiv":1,"publication_identifier":{"isbn":["9789819571260"],"issn":["0302-9743"],"eissn":["1611-3349"]},"title":"On the MST-ratio: Theoretical bounds and complexity of finding the maximum","date_created":"2026-03-08T23:01:45Z","alternative_title":["LNCS"],"oa_version":"Preprint","OA_type":"green","conference":{"name":"WALCOM: International Conference and Workshops on Algorithms and Computation","start_date":"2026-03-04","end_date":"2026-03-06","location":"Perugia, Italy"},"publication_status":"published"},{"isi":1,"date_updated":"2025-04-15T07:16:53Z","publication":"Advances in Mathematics","scopus_import":"1","ec_funded":1,"status":"public","day":"01","citation":{"ama":"Edelsbrunner H, Garber A, Saghafian M. Order-2 Delaunay triangulations optimize angles. <i>Advances in Mathematics</i>. 2025;461. doi:<a href=\"https://doi.org/10.1016/j.aim.2024.110055\">10.1016/j.aim.2024.110055</a>","ista":"Edelsbrunner H, Garber A, Saghafian M. 2025. Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. 461, 110055.","chicago":"Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “Order-2 Delaunay Triangulations Optimize Angles.” <i>Advances in Mathematics</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.aim.2024.110055\">https://doi.org/10.1016/j.aim.2024.110055</a>.","mla":"Edelsbrunner, Herbert, et al. “Order-2 Delaunay Triangulations Optimize Angles.” <i>Advances in Mathematics</i>, vol. 461, 110055, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.aim.2024.110055\">10.1016/j.aim.2024.110055</a>.","short":"H. Edelsbrunner, A. Garber, M. Saghafian, Advances in Mathematics 461 (2025).","ieee":"H. Edelsbrunner, A. Garber, and M. Saghafian, “Order-2 Delaunay triangulations optimize angles,” <i>Advances in Mathematics</i>, vol. 461. Elsevier, 2025.","apa":"Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). Order-2 Delaunay triangulations optimize angles. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2024.110055\">https://doi.org/10.1016/j.aim.2024.110055</a>"},"type":"journal_article","quality_controlled":"1","_id":"18626","language":[{"iso":"eng"}],"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."}],"external_id":{"isi":["001370682500001"],"arxiv":["2310.18238"]},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2310.18238"}],"date_published":"2025-02-01T00:00:00Z","OA_place":"repository","year":"2025","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":"       461","article_number":"110055","article_type":"original","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"name":"Mathematics, Computer Science","grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"month":"02","department":[{"_id":"HeEd"}],"author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"full_name":"Garber, Alexey","last_name":"Garber","first_name":"Alexey"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"volume":461,"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.","doi":"10.1016/j.aim.2024.110055","oa":1,"publisher":"Elsevier","corr_author":"1","article_processing_charge":"No","publication_status":"published","oa_version":"Preprint","OA_type":"green","date_created":"2024-12-08T23:01:54Z","title":"Order-2 Delaunay triangulations optimize angles","arxiv":1,"publication_identifier":{"issn":["0001-8708"],"eissn":["1090-2082"]}},{"isi":1,"date_updated":"2025-12-30T09:05:32Z","publication":"Information Sciences","scopus_import":"1","issue":"11","ec_funded":1,"status":"public","citation":{"apa":"Mahini, M., Beigy, H., Qadami, S., &#38; Saghafian, M. (2025). Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. <i>Information Sciences</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.ins.2025.122425\">https://doi.org/10.1016/j.ins.2025.122425</a>","mla":"Mahini, Mohammad, et al. “Simplet-Based Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.” <i>Information Sciences</i>, vol. 719, no. 11, 122425, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.ins.2025.122425\">10.1016/j.ins.2025.122425</a>.","short":"M. Mahini, H. Beigy, S. Qadami, M. Saghafian, Information Sciences 719 (2025).","ieee":"M. Mahini, H. Beigy, S. Qadami, and M. Saghafian, “Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality,” <i>Information Sciences</i>, vol. 719, no. 11. Elsevier, 2025.","chicago":"Mahini, Mohammad, Hamid Beigy, Salman Qadami, and Morteza Saghafian. “Simplet-Based Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.” <i>Information Sciences</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.ins.2025.122425\">https://doi.org/10.1016/j.ins.2025.122425</a>.","ama":"Mahini M, Beigy H, Qadami S, Saghafian M. Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. <i>Information Sciences</i>. 2025;719(11). doi:<a href=\"https://doi.org/10.1016/j.ins.2025.122425\">10.1016/j.ins.2025.122425</a>","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."},"day":"01","quality_controlled":"1","type":"journal_article","external_id":{"isi":["001516170500002"]},"_id":"19937","language":[{"iso":"eng"}],"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."}],"date_published":"2025-11-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2025","intvolume":"       719","article_number":"122425","month":"11","department":[{"_id":"HeEd"}],"article_type":"original","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","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"}],"author":[{"first_name":"Mohammad","full_name":"Mahini, Mohammad","last_name":"Mahini"},{"full_name":"Beigy, Hamid","last_name":"Beigy","first_name":"Hamid"},{"first_name":"Salman","last_name":"Qadami","full_name":"Qadami, Salman"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"volume":719,"doi":"10.1016/j.ins.2025.122425","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.","publisher":"Elsevier","article_processing_charge":"No","corr_author":"1","publication_status":"published","OA_type":"closed access","oa_version":"None","title":"Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality","date_created":"2025-06-30T08:48:48Z","publication_identifier":{"issn":["0020-0255"]}},{"corr_author":"1","tmp":{"short":"CC BY (4.0)","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)"},"article_processing_charge":"Yes","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","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","doi":"10.4230/LIPIcs.SoCG.2025.43","oa":1,"volume":332,"author":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Alexey","full_name":"Garber, Alexey","last_name":"Garber"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"project":[{"name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"department":[{"_id":"HeEd"}],"month":"06","publication_identifier":{"eissn":["1868-8969"],"isbn":["9783959773706"]},"arxiv":1,"title":"On spheres with k points inside","date_created":"2025-07-13T22:01:22Z","alternative_title":["LIPIcs"],"oa_version":"Published Version","OA_type":"gold","publication_status":"published","conference":{"name":"SoCG: Symposium on Computational Geometry","start_date":"2025-06-23","end_date":"2025-06-27","location":"Kanazawa, Japan"},"file":[{"access_level":"open_access","content_type":"application/pdf","success":1,"date_created":"2025-07-14T07:24:22Z","file_size":661893,"checksum":"b5313ed8575ea87913c71a6e3c7513c8","file_name":"2025_LIPIcs.SoCG_Edelsbrunner.pdf","relation":"main_file","date_updated":"2025-07-14T07:24:22Z","file_id":"20016","creator":"dernst"}],"status":"public","scopus_import":"1","file_date_updated":"2025-07-14T07:24:22Z","publication":"41st International Symposium on Computational Geometry","date_updated":"2025-07-14T07:26:14Z","intvolume":"       332","article_number":"43","year":"2025","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ddc":["510"],"date_published":"2025-06-20T00:00:00Z","OA_place":"publisher","language":[{"iso":"eng"}],"_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 𝕊^d. Furthermore, we use the result to give a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers."}],"external_id":{"arxiv":["2410.21204"]},"type":"conference","quality_controlled":"1","has_accepted_license":"1","citation":{"short":"H. Edelsbrunner, A. Garber, M. Saghafian, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","ieee":"H. Edelsbrunner, A. Garber, and M. Saghafian, “On spheres with k points inside,” in <i>41st International Symposium on Computational Geometry</i>, Kanazawa, Japan, 2025, vol. 332.","mla":"Edelsbrunner, Herbert, et al. “On Spheres with k Points Inside.” <i>41st International Symposium on Computational Geometry</i>, vol. 332, 43, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">10.4230/LIPIcs.SoCG.2025.43</a>.","apa":"Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). On spheres with k points inside. In <i>41st International Symposium on Computational Geometry</i> (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>","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.","ama":"Edelsbrunner H, Garber A, Saghafian M. On spheres with k points inside. In: <i>41st International Symposium on Computational Geometry</i>. Vol 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">10.4230/LIPIcs.SoCG.2025.43</a>","chicago":"Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “On Spheres with k Points Inside.” In <i>41st International Symposium on Computational Geometry</i>, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>."},"day":"20"},{"status":"public","ec_funded":1,"scopus_import":"1","publication":"European Journal of Combinatorics","date_updated":"2025-12-01T12:57:29Z","isi":1,"article_number":"104248","intvolume":"       132","year":"2025","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2025-10-10T00:00:00Z","OA_place":"repository","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2212.11380","open_access":"1"}],"_id":"20490","language":[{"iso":"eng"}],"abstract":[{"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","lang":"eng"}],"external_id":{"isi":["001599061500002"],"arxiv":["2212.11380"]},"type":"journal_article","quality_controlled":"1","day":"10","citation":{"apa":"Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., &#38; Saghafian, M. (2025). Flips in two-dimensional hypertriangulations. <i>European Journal of Combinatorics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">https://doi.org/10.1016/j.ejc.2025.104248</a>","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, European Journal of Combinatorics 132 (2025).","ieee":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “Flips in two-dimensional hypertriangulations,” <i>European Journal of Combinatorics</i>, vol. 132. Elsevier, 2025.","mla":"Edelsbrunner, Herbert, et al. “Flips in Two-Dimensional Hypertriangulations.” <i>European Journal of Combinatorics</i>, vol. 132, 104248, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">10.1016/j.ejc.2025.104248</a>.","chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “Flips in Two-Dimensional Hypertriangulations.” <i>European Journal of Combinatorics</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">https://doi.org/10.1016/j.ejc.2025.104248</a>.","ista":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2025. Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. 132, 104248.","ama":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. Flips in two-dimensional hypertriangulations. <i>European Journal of Combinatorics</i>. 2025;132. doi:<a href=\"https://doi.org/10.1016/j.ejc.2025.104248\">10.1016/j.ejc.2025.104248</a>"},"corr_author":"1","article_processing_charge":"No","publisher":"Elsevier","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.","doi":"10.1016/j.ejc.2025.104248","oa":1,"author":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Garber, Alexey","last_name":"Garber","first_name":"Alexey"},{"full_name":"Ghafari, Mohadese","last_name":"Ghafari","first_name":"Mohadese"},{"orcid":"0000-0002-1780-2689","first_name":"Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","last_name":"Heiss","full_name":"Heiss, Teresa"},{"first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza","last_name":"Saghafian"}],"volume":132,"article_type":"original","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Mathematics, Computer Science"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"month":"10","department":[{"_id":"HeEd"}],"publication_identifier":{"issn":["0195-6698"]},"arxiv":1,"date_created":"2025-10-19T22:01:31Z","title":"Flips in two-dimensional hypertriangulations","oa_version":"Preprint","OA_type":"green","publication_status":"epub_ahead"},{"publication":"Foundations of Data Science","date_updated":"2025-11-04T12:25:47Z","page":"30-62","status":"public","ec_funded":1,"scopus_import":"1","external_id":{"arxiv":["2212.03128"]},"_id":"20585","language":[{"iso":"eng"}],"abstract":[{"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.","lang":"eng"}],"quality_controlled":"1","type":"journal_article","citation":{"apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2025). Chromatic alpha complexes. <i>Foundations of Data Science</i>. American Institute of Mathematical Sciences. <a href=\"https://doi.org/10.3934/fods.2025003\">https://doi.org/10.3934/fods.2025003</a>","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic alpha complexes,” <i>Foundations of Data Science</i>, vol. 8. American Institute of Mathematical Sciences, pp. 30–62, 2025.","short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Foundations of Data Science 8 (2025) 30–62.","mla":"Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” <i>Foundations of Data Science</i>, vol. 8, American Institute of Mathematical Sciences, 2025, pp. 30–62, doi:<a href=\"https://doi.org/10.3934/fods.2025003\">10.3934/fods.2025003</a>.","chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Alpha Complexes.” <i>Foundations of Data Science</i>. American Institute of Mathematical Sciences, 2025. <a href=\"https://doi.org/10.3934/fods.2025003\">https://doi.org/10.3934/fods.2025003</a>.","ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2025. Chromatic alpha complexes. Foundations of Data Science. 8, 30–62.","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. <i>Foundations of Data Science</i>. 2025;8:30-62. doi:<a href=\"https://doi.org/10.3934/fods.2025003\">10.3934/fods.2025003</a>"},"day":"01","intvolume":"         8","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2025","OA_place":"repository","date_published":"2025-03-01T00:00:00Z","doi":"10.3934/fods.2025003","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.","author":[{"full_name":"Cultrera di Montesano, Sebastiano","last_name":"Cultrera di Montesano","orcid":"0000-0001-6249-0832","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano"},{"full_name":"Draganov, Ondrej","last_name":"Draganov","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","first_name":"Ondrej","orcid":"0000-0003-0464-3823"},{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"volume":8,"department":[{"_id":"HeEd"}],"month":"03","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Persistence and stability of geometric complexes"}],"article_type":"original","article_processing_charge":"No","corr_author":"1","publisher":"American Institute of Mathematical Sciences","related_material":{"record":[{"relation":"earlier_version","status":"public","id":"15091"}]},"publication_status":"epub_ahead","publication_identifier":{"eissn":["2639-8001"]},"arxiv":1,"title":"Chromatic alpha complexes","date_created":"2025-11-02T23:01:33Z","OA_type":"green","oa_version":"Preprint"},{"publication_identifier":{"issn":["0925-7721"]},"arxiv":1,"OA_type":"green","oa_version":"Preprint","date_created":"2026-02-16T15:48:42Z","title":"Decomposition of geometric graphs into star-forests","publication_status":"published","publisher":"Elsevier","article_processing_charge":"No","corr_author":"1","related_material":{"record":[{"relation":"earlier_version","id":"15012","status":"public"}]},"oa":1,"doi":"10.1016/j.comgeo.2025.102186","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.","department":[{"_id":"HeEd"}],"month":"12","article_type":"original","volume":129,"author":[{"first_name":"János","full_name":"Pach, János","last_name":"Pach"},{"first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza","last_name":"Saghafian"},{"full_name":"Schnider, Patrick","last_name":"Schnider","first_name":"Patrick"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2025","article_number":"102186","intvolume":"       129","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2306.13201","open_access":"1"}],"OA_place":"repository","date_published":"2025-12-01T00:00:00Z","quality_controlled":"1","type":"journal_article","external_id":{"arxiv":["2306.13201"]},"_id":"21253","abstract":[{"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.","lang":"eng"}],"language":[{"iso":"eng"}],"citation":{"ama":"Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests. <i>Computational Geometry</i>. 2025;129. doi:<a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">10.1016/j.comgeo.2025.102186</a>","ista":"Pach J, Saghafian M, Schnider P. 2025. Decomposition of geometric graphs into star-forests. Computational Geometry. 129, 102186.","chicago":"Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric Graphs into Star-Forests.” <i>Computational Geometry</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">https://doi.org/10.1016/j.comgeo.2025.102186</a>.","mla":"Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.” <i>Computational Geometry</i>, vol. 129, 102186, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">10.1016/j.comgeo.2025.102186</a>.","short":"J. Pach, M. Saghafian, P. Schnider, Computational Geometry 129 (2025).","ieee":"J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs into star-forests,” <i>Computational Geometry</i>, vol. 129. Elsevier, 2025.","apa":"Pach, J., Saghafian, M., &#38; Schnider, P. (2025). Decomposition of geometric graphs into star-forests. <i>Computational Geometry</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">https://doi.org/10.1016/j.comgeo.2025.102186</a>"},"day":"01","status":"public","publication":"Computational Geometry","date_updated":"2026-04-16T09:12:36Z"},{"abstract":[{"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)).","lang":"eng"}],"_id":"14345","language":[{"iso":"eng"}],"external_id":{"pmid":["39610762"],"isi":["001060727600004"],"arxiv":["2204.01076"]},"type":"journal_article","quality_controlled":"1","has_accepted_license":"1","citation":{"mla":"Edelsbrunner, Herbert, et al. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>, vol. 72, Springer Nature, 2024, pp. 29–48, doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>.","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete and Computational Geometry 72 (2024) 29–48.","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,” <i>Discrete and Computational Geometry</i>, vol. 72. Springer Nature, pp. 29–48, 2024.","apa":"Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., &#38; Saghafian, M. (2024). On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>","ama":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. 2024;72:29-48. doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>","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.","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.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>."},"day":"01","intvolume":"        72","year":"2024","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ddc":["510"],"date_published":"2024-07-01T00:00:00Z","file_date_updated":"2024-07-22T09:43:19Z","publication":"Discrete and Computational Geometry","date_updated":"2025-04-23T08:41:59Z","isi":1,"page":"29-48","status":"public","ec_funded":1,"scopus_import":"1","publication_status":"published","file":[{"date_updated":"2024-07-22T09:43:19Z","relation":"main_file","creator":"dernst","file_id":"17301","checksum":"b207b4e00f904e8ea8a30e24f0251f79","file_name":"2024_DiscreteComputGeom_Edelsbrunner.pdf","content_type":"application/pdf","success":1,"date_created":"2024-07-22T09:43:19Z","file_size":892019,"access_level":"open_access"}],"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"arxiv":1,"title":"On angles in higher order Brillouin tessellations and related tilings in the plane","date_created":"2023-09-17T22:01:10Z","oa_version":"Published Version","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.","oa":1,"doi":"10.1007/s00454-023-00566-1","author":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833"},{"last_name":"Garber","full_name":"Garber, Alexey","first_name":"Alexey"},{"last_name":"Ghafari","full_name":"Ghafari, Mohadese","first_name":"Mohadese"},{"full_name":"Heiss, Teresa","last_name":"Heiss","orcid":"0000-0002-1780-2689","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","first_name":"Teresa"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"volume":72,"article_type":"original","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"},{"name":"Mathematics, Computer Science","grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35"}],"department":[{"_id":"HeEd"}],"month":"07","corr_author":"1","tmp":{"short":"CC BY (4.0)","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)"},"article_processing_charge":"Yes (via OA deal)","publisher":"Springer Nature","pmid":1},{"type":"conference","quality_controlled":"1","_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."}],"language":[{"iso":"eng"}],"external_id":{"isi":["001540278400001"],"arxiv":["2403.10204"]},"has_accepted_license":"1","citation":{"chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “The Euclidean MST-Ratio for Bi-Colored Lattices.” In <i>32nd International Symposium on Graph Drawing and Network Visualization</i>, Vol. 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.GD.2024.3\">https://doi.org/10.4230/LIPIcs.GD.2024.3</a>.","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. The Euclidean MST-ratio for bi-colored lattices. In: <i>32nd International Symposium on Graph Drawing and Network Visualization</i>. Vol 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.GD.2024.3\">10.4230/LIPIcs.GD.2024.3</a>","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.","apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2024). The Euclidean MST-ratio for bi-colored lattices. In <i>32nd International Symposium on Graph Drawing and Network Visualization</i> (Vol. 320). Vienna, Austria: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.GD.2024.3\">https://doi.org/10.4230/LIPIcs.GD.2024.3</a>","mla":"Cultrera di Montesano, Sebastiano, et al. “The Euclidean MST-Ratio for Bi-Colored Lattices.” <i>32nd International Symposium on Graph Drawing and Network Visualization</i>, vol. 320, 3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.GD.2024.3\">10.4230/LIPIcs.GD.2024.3</a>.","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.","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “The Euclidean MST-ratio for bi-colored lattices,” in <i>32nd International Symposium on Graph Drawing and Network Visualization</i>, Vienna, Austria, 2024, vol. 320."},"day":"28","year":"2024","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ddc":["510"],"intvolume":"       320","article_number":"3","date_published":"2024-10-28T00:00:00Z","OA_place":"publisher","file_date_updated":"2024-11-18T07:49:25Z","isi":1,"publication":"32nd International Symposium on Graph Drawing and Network Visualization","date_updated":"2025-12-02T13:50:50Z","status":"public","scopus_import":"1","ec_funded":1,"publication_status":"published","conference":{"location":"Vienna, Austria","end_date":"2024-09-20","start_date":"2024-09-18","name":"GD: Graph Drawing and Network Visualization"},"file":[{"date_created":"2024-11-18T07:49:25Z","file_size":908541,"content_type":"application/pdf","success":1,"access_level":"open_access","date_updated":"2024-11-18T07:49:25Z","file_id":"18560","relation":"main_file","creator":"dernst","file_name":"2024_LIPIcs_CultreradiMontesano.pdf","checksum":"5f9b35e115c3d375e99be78da9054cb4"}],"publication_identifier":{"issn":["1868-8969"],"isbn":["9783959773430"]},"arxiv":1,"oa_version":"Published Version","OA_type":"gold","title":"The Euclidean MST-ratio for bi-colored lattices","alternative_title":["LIPIcs"],"date_created":"2024-11-17T23:01:47Z","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.","doi":"10.4230/LIPIcs.GD.2024.3","oa":1,"project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"Mathematics, Computer Science"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"department":[{"_id":"HeEd"}],"month":"10","author":[{"last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832"},{"first_name":"Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0464-3823","full_name":"Draganov, Ondrej","last_name":"Draganov"},{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833"},{"first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza","last_name":"Saghafian"}],"volume":320,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","tmp":{"short":"CC BY (4.0)","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)"},"corr_author":"1","article_processing_charge":"Yes"},{"external_id":{"arxiv":["2406.04102"]},"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."}],"_id":"18999","language":[{"iso":"eng"}],"type":"preprint","day":"06","citation":{"apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (n.d.). Chromatic topological data analysis. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/ARXIV.2406.04102\">https://doi.org/10.48550/ARXIV.2406.04102</a>","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic topological data analysis,” <i>arXiv</i>. .","short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.).","mla":"Cultrera di Montesano, Sebastiano, et al. “Chromatic Topological Data Analysis.” <i>ArXiv</i>, 2406.04102, doi:<a href=\"https://doi.org/10.48550/ARXIV.2406.04102\">10.48550/ARXIV.2406.04102</a>.","chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Topological Data Analysis.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/ARXIV.2406.04102\">https://doi.org/10.48550/ARXIV.2406.04102</a>.","ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic topological data analysis. arXiv, 2406.04102.","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic topological data analysis. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/ARXIV.2406.04102\">10.48550/ARXIV.2406.04102</a>"},"has_accepted_license":"1","article_number":"2406.04102","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ddc":["510"],"year":"2024","OA_place":"repository","date_published":"2024-06-06T00:00:00Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2406.04102","open_access":"1"}],"date_updated":"2025-02-10T08:14:27Z","publication":"arXiv","status":"public","publication_status":"submitted","arxiv":1,"date_created":"2025-02-04T16:21:21Z","title":"Chromatic topological data analysis","OA_type":"green","oa_version":"Preprint","oa":1,"doi":"10.48550/ARXIV.2406.04102","author":[{"full_name":"Cultrera di Montesano, Sebastiano","last_name":"Cultrera di Montesano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","first_name":"Sebastiano","orcid":"0000-0001-6249-0832"},{"full_name":"Draganov, Ondrej","last_name":"Draganov","orcid":"0000-0003-0464-3823","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","first_name":"Ondrej"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian","full_name":"Saghafian, Morteza"}],"department":[{"_id":"GradSch"},{"_id":"HeEd"}],"month":"06","article_processing_charge":"No","tmp":{"short":"CC BY (4.0)","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)"},"corr_author":"1"},{"date_updated":"2025-05-14T09:27:57Z","publication":"Journal of Applied and Computational Topology","file_date_updated":"2025-04-23T08:01:36Z","ec_funded":1,"scopus_import":"1","page":"557-578","status":"public","day":"01","citation":{"mla":"Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Journal of Applied and Computational Topology</i>, vol. 8, Springer Nature, 2024, pp. 557–78, doi:<a href=\"https://doi.org/10.1007/s41468-024-00173-w\">10.1007/s41468-024-00173-w</a>.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology 8 (2024) 557–578.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth in arrangements: Dehn–Sommerville–Euler relations with applications,” <i>Journal of Applied and Computational Topology</i>, vol. 8. Springer Nature, pp. 557–578, 2024.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2024). Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-024-00173-w\">https://doi.org/10.1007/s41468-024-00173-w</a>","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Journal of Applied and Computational Topology</i>. 2024;8:557-578. doi:<a href=\"https://doi.org/10.1007/s41468-024-00173-w\">10.1007/s41468-024-00173-w</a>","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.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s41468-024-00173-w\">https://doi.org/10.1007/s41468-024-00173-w</a>."},"has_accepted_license":"1","external_id":{"pmid":["39308789"]},"_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."}],"language":[{"iso":"eng"}],"quality_controlled":"1","type":"journal_article","OA_place":"publisher","date_published":"2024-09-01T00:00:00Z","intvolume":"         8","ddc":["510"],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2024","author":[{"orcid":"0000-0002-5372-7890","first_name":"Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","full_name":"Biswas, Ranita","last_name":"Biswas"},{"full_name":"Cultrera Di Montesano, Sebastiano","last_name":"Cultrera Di Montesano","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832"},{"orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian","full_name":"Saghafian, Morteza"}],"volume":8,"month":"09","department":[{"_id":"HeEd"}],"article_type":"original","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Mathematics, Computer Science"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"doi":"10.1007/s41468-024-00173-w","oa":1,"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.","related_material":{"record":[{"relation":"earlier_version","id":"11658","status":"public"}]},"pmid":1,"article_processing_charge":"Yes (via OA deal)","corr_author":"1","tmp":{"short":"CC BY (4.0)","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)"},"publisher":"Springer Nature","file":[{"relation":"main_file","file_id":"19612","creator":"dernst","date_updated":"2025-04-23T08:01:36Z","file_name":"2024_JourApplCompTopo_BiswasRa.pdf","checksum":"0ee15c1493a6413cf356ab2f32c81a9e","date_created":"2025-04-23T08:01:36Z","file_size":522831,"content_type":"application/pdf","success":1,"access_level":"open_access"}],"publication_status":"published","title":"Depth in arrangements: Dehn–Sommerville–Euler relations with applications","date_created":"2024-05-12T22:01:03Z","OA_type":"hybrid","oa_version":"Published Version","publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]}},{"year":"2024","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ddc":["510"],"intvolume":"       293","article_number":"76","date_published":"2024-06-01T00:00:00Z","type":"conference","quality_controlled":"1","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."}],"_id":"17145","language":[{"iso":"eng"}],"external_id":{"arxiv":["2402.15787"]},"has_accepted_license":"1","citation":{"chicago":"Rote, Günter, Moritz Rüber, and Morteza Saghafian. “Grid Peeling of Parabolas.” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.76\">https://doi.org/10.4230/LIPIcs.SoCG.2024.76</a>.","ama":"Rote G, Rüber M, Saghafian M. Grid peeling of parabolas. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.76\">10.4230/LIPIcs.SoCG.2024.76</a>","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.","apa":"Rote, G., Rüber, M., &#38; Saghafian, M. (2024). Grid peeling of parabolas. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.76\">https://doi.org/10.4230/LIPIcs.SoCG.2024.76</a>","mla":"Rote, Günter, et al. “Grid Peeling of Parabolas.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 76, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.76\">10.4230/LIPIcs.SoCG.2024.76</a>.","ieee":"G. Rote, M. Rüber, and M. Saghafian, “Grid peeling of parabolas,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","short":"G. Rote, M. Rüber, M. Saghafian, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024."},"day":"01","status":"public","scopus_import":"1","file_date_updated":"2024-06-17T08:40:04Z","date_updated":"2024-06-17T08:41:56Z","publication":"40th International Symposium on Computational Geometry","publication_identifier":{"isbn":["9783959773164"],"issn":["1868-8969"]},"arxiv":1,"oa_version":"Published Version","title":"Grid peeling of parabolas","date_created":"2024-06-16T22:01:06Z","alternative_title":["LIPIcs"],"publication_status":"published","conference":{"name":"SoCG: Symposium on Computational Geometry","start_date":"2024-06-11","end_date":"2024-06-14","location":"Athens, Greece"},"file":[{"file_name":"2024_LIPICS_Rote.pdf","checksum":"fbad1de06383a6b7e8a1cb3e8c7205ce","file_id":"17151","date_updated":"2024-06-17T08:40:04Z","creator":"dernst","relation":"main_file","access_level":"open_access","date_created":"2024-06-17T08:40:04Z","file_size":1430896,"content_type":"application/pdf","success":1}],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","tmp":{"short":"CC BY (4.0)","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)"},"article_processing_charge":"No","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.","doi":"10.4230/LIPIcs.SoCG.2024.76","oa":1,"month":"06","department":[{"_id":"HeEd"}],"volume":293,"author":[{"last_name":"Rote","full_name":"Rote, Günter","first_name":"Günter"},{"first_name":"Moritz","last_name":"Rüber","full_name":"Rüber, Moritz"},{"id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza","full_name":"Saghafian, Morteza","last_name":"Saghafian"}]},{"abstract":[{"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.","lang":"eng"}],"_id":"13182","language":[{"iso":"eng"}],"external_id":{"pmid":["39678706"]},"type":"journal_article","quality_controlled":"1","has_accepted_license":"1","citation":{"ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Geometric characterization of the persistence of 1D maps,” <i>Journal of Applied and Computational Topology</i>, vol. 8. Springer Nature, pp. 1101–1119, 2024.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology 8 (2024) 1101–1119.","mla":"Biswas, Ranita, et al. “Geometric Characterization of the Persistence of 1D Maps.” <i>Journal of Applied and Computational Topology</i>, vol. 8, Springer Nature, 2024, pp. 1101–19, doi:<a href=\"https://doi.org/10.1007/s41468-023-00126-9\">10.1007/s41468-023-00126-9</a>.","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2024). Geometric characterization of the persistence of 1D maps. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-023-00126-9\">https://doi.org/10.1007/s41468-023-00126-9</a>","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.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Geometric characterization of the persistence of 1D maps. <i>Journal of Applied and Computational Topology</i>. 2024;8:1101-1119. doi:<a href=\"https://doi.org/10.1007/s41468-023-00126-9\">10.1007/s41468-023-00126-9</a>","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Geometric Characterization of the Persistence of 1D Maps.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s41468-023-00126-9\">https://doi.org/10.1007/s41468-023-00126-9</a>."},"day":"01","intvolume":"         8","year":"2024","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ddc":["000"],"date_published":"2024-10-01T00:00:00Z","OA_place":"publisher","file_date_updated":"2025-01-09T07:39:41Z","date_updated":"2026-04-07T12:58:47Z","publication":"Journal of Applied and Computational Topology","page":"1101-1119","status":"public","ec_funded":1,"scopus_import":"1","publication_status":"published","file":[{"success":1,"content_type":"application/pdf","date_created":"2025-01-09T07:39:41Z","file_size":476896,"access_level":"open_access","file_id":"18783","relation":"main_file","creator":"dernst","date_updated":"2025-01-09T07:39:41Z","checksum":"d493df5088c222b88d9ca46b623ad0ee","file_name":"2024_JourApplCompTopo_Biswas.pdf"}],"publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]},"date_created":"2023-07-02T22:00:44Z","title":"Geometric characterization of the persistence of 1D maps","oa_version":"Published Version","OA_type":"hybrid","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.","doi":"10.1007/s41468-023-00126-9","oa":1,"author":[{"last_name":"Biswas","full_name":"Biswas, Ranita","first_name":"Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-5372-7890"},{"full_name":"Cultrera Di Montesano, Sebastiano","last_name":"Cultrera Di Montesano","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832"},{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza","last_name":"Saghafian"}],"volume":8,"article_type":"original","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"name":"Persistent Homology, Algorithms and Stochastic Geometry","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","grant_number":"I4887"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342","name":"Mathematics, Computer Science"}],"month":"10","department":[{"_id":"HeEd"}],"tmp":{"short":"CC BY (4.0)","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)"},"corr_author":"1","article_processing_charge":"Yes (via OA deal)","publisher":"Springer Nature","pmid":1,"related_material":{"record":[{"relation":"dissertation_contains","id":"15094","status":"public"}]}},{"oa":1,"doi":"10.48550/arXiv.2212.03128","date_updated":"2026-04-07T12:58:47Z","publication":"arXiv","author":[{"last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","orcid":"0000-0001-6249-0832","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0003-0464-3823","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","first_name":"Ondrej","full_name":"Draganov, Ondrej","last_name":"Draganov"},{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"department":[{"_id":"HeEd"}],"month":"02","corr_author":"1","tmp":{"short":"CC BY (4.0)","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)"},"article_processing_charge":"No","status":"public","related_material":{"record":[{"id":"20585","status":"public","relation":"later_version"},{"relation":"dissertation_contains","status":"public","id":"18979"},{"relation":"dissertation_contains","status":"public","id":"15094"}]},"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."}],"_id":"15091","language":[{"iso":"eng"}],"external_id":{"arxiv":["2212.03128"]},"type":"preprint","publication_status":"draft","day":"07","citation":{"mla":"Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” <i>ArXiv</i>, 2212.03128, doi:<a href=\"https://doi.org/10.48550/arXiv.2212.03128\">10.48550/arXiv.2212.03128</a>.","short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.).","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic alpha complexes,” <i>arXiv</i>. .","apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (n.d.). Chromatic alpha complexes. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2212.03128\">https://doi.org/10.48550/arXiv.2212.03128</a>","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2212.03128\">10.48550/arXiv.2212.03128</a>","ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. arXiv, 2212.03128.","chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Alpha Complexes.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2212.03128\">https://doi.org/10.48550/arXiv.2212.03128</a>."},"article_number":"2212.03128","arxiv":1,"year":"2024","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","date_published":"2024-02-07T00:00:00Z","date_created":"2024-03-08T10:13:59Z","title":"Chromatic alpha complexes","OA_place":"repository","oa_version":"Preprint","main_file_link":[{"url":"https://arxiv.org/abs/2212.03128","open_access":"1"}]},{"date_updated":"2026-04-16T09:12:37Z","publication":"31st International Symposium on Graph Drawing and Network Visualization","isi":1,"status":"public","page":"339-346","ec_funded":1,"scopus_import":"1","external_id":{"isi":["001207939600023"],"arxiv":["2306.13201"]},"language":[{"iso":"eng"}],"_id":"15012","abstract":[{"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.","lang":"eng"}],"quality_controlled":"1","type":"conference","day":"01","citation":{"apa":"Pach, J., Saghafian, M., &#38; Schnider, P. (2024). Decomposition of geometric graphs into star-forests. In <i>31st International Symposium on Graph Drawing and Network Visualization</i> (Vol. 14465, pp. 339–346). Isola delle Femmine, Palermo, Italy: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">https://doi.org/10.1007/978-3-031-49272-3_23</a>","mla":"Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.” <i>31st International Symposium on Graph Drawing and Network Visualization</i>, vol. 14465, Springer Nature, 2024, pp. 339–46, doi:<a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">10.1007/978-3-031-49272-3_23</a>.","ieee":"J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs into star-forests,” in <i>31st International Symposium on Graph Drawing and Network Visualization</i>, Isola delle Femmine, Palermo, Italy, 2024, vol. 14465, pp. 339–346.","short":"J. Pach, M. Saghafian, P. Schnider, in:, 31st International Symposium on Graph Drawing and Network Visualization, Springer Nature, 2024, pp. 339–346.","chicago":"Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric Graphs into Star-Forests.” In <i>31st International Symposium on Graph Drawing and Network Visualization</i>, 14465:339–46. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">https://doi.org/10.1007/978-3-031-49272-3_23</a>.","ama":"Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests. In: <i>31st International Symposium on Graph Drawing and Network Visualization</i>. Vol 14465. Springer Nature; 2024:339-346. doi:<a href=\"https://doi.org/10.1007/978-3-031-49272-3_23\">10.1007/978-3-031-49272-3_23</a>","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."},"intvolume":"     14465","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","year":"2024","date_published":"2024-01-01T00:00:00Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2306.13201","open_access":"1"}],"oa":1,"doi":"10.1007/978-3-031-49272-3_23","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.","author":[{"full_name":"Pach, János","last_name":"Pach","first_name":"János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"},{"first_name":"Patrick","full_name":"Schnider, Patrick","last_name":"Schnider"}],"volume":14465,"department":[{"_id":"HeEd"}],"month":"01","project":[{"grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"Mathematics, Computer Science"}],"article_processing_charge":"No","publisher":"Springer Nature","related_material":{"record":[{"relation":"later_version","id":"21253","status":"public"}]},"publication_status":"published","conference":{"location":"Isola delle Femmine, Palermo, Italy","end_date":"2023-09-22","start_date":"2023-09-20","name":"GD: Graph Drawing and Network Visualization"},"publication_identifier":{"isbn":["9783031492716"],"eisbn":["9783031492723"],"issn":["0302-9743"],"eissn":["1611-3349"]},"arxiv":1,"alternative_title":["LNCS"],"date_created":"2024-02-18T23:01:03Z","title":"Decomposition of geometric graphs into star-forests","oa_version":"Preprint"},{"date_published":"2022-07-27T00:00:00Z","year":"2022","ddc":["510"],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","has_accepted_license":"1","day":"27","citation":{"apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (n.d.). Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Leibniz International Proceedings on Mathematics</i>. Schloss Dagstuhl - Leibniz-Zentrum für Informatik.","mla":"Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Leibniz International Proceedings on Mathematics</i>, Schloss Dagstuhl - Leibniz-Zentrum für Informatik.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth in arrangements: Dehn–Sommerville–Euler relations with applications,” <i>Leibniz International Proceedings on Mathematics</i>. Schloss Dagstuhl - Leibniz-Zentrum für Informatik.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Leibniz International Proceedings on Mathematics (n.d.).","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” <i>Leibniz International Proceedings on Mathematics</i>. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, n.d.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. <i>Leibniz International Proceedings on Mathematics</i>.","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."},"abstract":[{"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.","lang":"eng"}],"_id":"11658","language":[{"iso":"eng"}],"type":"journal_article","quality_controlled":"1","ec_funded":1,"status":"public","publication":"Leibniz International Proceedings on Mathematics","date_updated":"2026-04-07T12:58:48Z","file_date_updated":"2022-07-27T09:25:53Z","date_created":"2022-07-27T09:27:34Z","title":"Depth in arrangements: Dehn–Sommerville–Euler relations with applications","oa_version":"Submitted Version","file":[{"file_id":"11659","relation":"main_file","creator":"scultrer","date_updated":"2022-07-27T09:25:53Z","checksum":"b2f511e8b1cae5f1892b0cdec341acac","file_name":"D-S-E.pdf","content_type":"application/pdf","file_size":639266,"date_created":"2022-07-27T09:25:53Z","access_level":"open_access"}],"publication_status":"draft","related_material":{"record":[{"id":"15380","status":"public","relation":"later_version"},{"status":"public","id":"15094","relation":"dissertation_contains"}]},"tmp":{"short":"CC BY (4.0)","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)"},"corr_author":"1","article_processing_charge":"No","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","author":[{"orcid":"0000-0002-5372-7890","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","first_name":"Ranita","last_name":"Biswas","full_name":"Biswas, Ranita"},{"orcid":"0000-0001-6249-0832","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","full_name":"Cultrera di Montesano, Sebastiano","last_name":"Cultrera di Montesano"},{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"project":[{"name":"Alpha Shape Theory Extended","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"department":[{"_id":"GradSch"},{"_id":"HeEd"}],"month":"07","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.","oa":1},{"publication":"arXiv","date_updated":"2026-04-07T12:58:47Z","status":"public","ec_funded":1,"external_id":{"arxiv":["2212.03121"]},"_id":"15090","language":[{"iso":"eng"}],"abstract":[{"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.","lang":"eng"}],"type":"preprint","citation":{"apa":"Biswas, R., Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (n.d.). On the size of chromatic Delaunay mosaics. <i>arXiv</i>.","ieee":"R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “On the size of chromatic Delaunay mosaics,” <i>arXiv</i>. .","short":"R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.).","mla":"Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” <i>ArXiv</i>, 2212.03121.","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “On the Size of Chromatic Delaunay Mosaics.” <i>ArXiv</i>, n.d.","ista":"Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. arXiv, 2212.03121.","ama":"Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. <i>arXiv</i>."},"day":"06","article_number":"2212.03121","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2022","OA_place":"repository","date_published":"2022-12-06T00:00:00Z","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2212.03121"}],"oa":1,"author":[{"full_name":"Biswas, Ranita","last_name":"Biswas","orcid":"0000-0002-5372-7890","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","first_name":"Ranita"},{"last_name":"Cultrera di Montesano","full_name":"Cultrera di Montesano, Sebastiano","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832"},{"last_name":"Draganov","full_name":"Draganov, Ondrej","orcid":"0000-0003-0464-3823","first_name":"Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"department":[{"_id":"HeEd"}],"month":"12","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"},{"grant_number":"I4887","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","name":"Persistent Homology, Algorithms and Stochastic Geometry"},{"name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342"}],"article_processing_charge":"No","tmp":{"short":"CC BY (4.0)","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)"},"corr_author":"1","related_material":{"record":[{"id":"20456","status":"public","relation":"later_version"},{"status":"public","id":"15094","relation":"dissertation_contains"}]},"publication_status":"draft","arxiv":1,"date_created":"2024-03-08T09:54:20Z","title":"On the size of chromatic Delaunay mosaics","oa_version":"Preprint"}]