[{"scopus_import":"1","intvolume":" 64","month":"03","abstract":[{"text":"Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups.","lang":"eng"}],"oa_version":"Published Version","ec_funded":1,"volume":64,"publication_status":"published","publication_identifier":{"eissn":["14320444"],"issn":["01795376"]},"language":[{"iso":"eng"}],"file":[{"relation":"main_file","access_level":"open_access","content_type":"application/pdf","success":1,"checksum":"f8cc96e497f00c38340b5dafe0cb91d7","file_id":"8786","creator":"dernst","file_size":701673,"date_updated":"2020-11-20T13:22:21Z","file_name":"2020_DiscreteCompGeo_Edelsbrunner.pdf","date_created":"2020-11-20T13:22:21Z"}],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"article_type":"original","type":"journal_article","status":"public","_id":"7666","file_date_updated":"2020-11-20T13:22:21Z","department":[{"_id":"HeEd"}],"date_updated":"2023-08-21T06:13:48Z","ddc":["510"],"oa":1,"publisher":"Springer Nature","quality_controlled":"1","acknowledgement":"This project has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through Grant No. I02979-N35 of the Austrian Science Fund (FWF).","page":"759-775","date_created":"2020-04-19T22:00:56Z","date_published":"2020-03-20T00:00:00Z","doi":"10.1007/s00454-020-00188-x","year":"2020","isi":1,"has_accepted_license":"1","publication":"Discrete and Computational Geometry","day":"20","project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"},{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"article_processing_charge":"Yes (via OA deal)","external_id":{"isi":["000520918800001"]},"author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","first_name":"Katharina","last_name":"Ölsböck","orcid":"0000-0002-4672-8297","full_name":"Ölsböck, Katharina"}],"title":"Tri-partitions and bases of an ordered complex","citation":{"ista":"Edelsbrunner H, Ölsböck K. 2020. Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. 64, 759–775.","chicago":"Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” Discrete and Computational Geometry. Springer Nature, 2020. https://doi.org/10.1007/s00454-020-00188-x.","ieee":"H. Edelsbrunner and K. Ölsböck, “Tri-partitions and bases of an ordered complex,” Discrete and Computational Geometry, vol. 64. Springer Nature, pp. 759–775, 2020.","short":"H. Edelsbrunner, K. Ölsböck, Discrete and Computational Geometry 64 (2020) 759–775.","apa":"Edelsbrunner, H., & Ölsböck, K. (2020). Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00188-x","ama":"Edelsbrunner H, Ölsböck K. Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. 2020;64:759-775. doi:10.1007/s00454-020-00188-x","mla":"Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” Discrete and Computational Geometry, vol. 64, Springer Nature, 2020, pp. 759–75, doi:10.1007/s00454-020-00188-x."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8"},{"abstract":[{"lang":"eng","text":"Let A={A1,…,An} be a family of sets in the plane. For 0≤i2b be integers. We prove that if each k-wise or (k+1)-wise intersection of sets from A has at most b path-connected components, which all are open, then fk+1=0 implies fk≤cfk−1 for some positive constant c depending only on b and k. These results also extend to two-dimensional compact surfaces."}],"oa_version":"Preprint","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1907.00885"}],"scopus_import":"1","intvolume":" 64","month":"09","publication_status":"published","publication_identifier":{"eissn":["14320444"],"issn":["01795376"]},"language":[{"iso":"eng"}],"volume":64,"_id":"7960","article_type":"original","type":"journal_article","status":"public","date_updated":"2023-08-21T08:26:34Z","department":[{"_id":"UlWa"}],"acknowledgement":"We are very grateful to Pavel Paták for many helpful discussions and remarks. We also thank the referees for helpful comments, which greatly improved the presentation.\r\nThe project was supported by ERC Advanced Grant 320924. GK was also partially supported by NSF grant DMS1300120. The research stay of ZP at IST Austria is funded by the project CZ.02.2.69/0.0/0.0/17_050/0008466 Improvement of internationalization in the field of research and development at Charles University, through the support of quality projects MSCA-IF.","oa":1,"quality_controlled":"1","publisher":"Springer Nature","year":"2020","isi":1,"publication":"Discrete and Computational Geometry","day":"01","page":"304-323","date_created":"2020-06-14T22:00:50Z","date_published":"2020-09-01T00:00:00Z","doi":"10.1007/s00454-020-00205-z","citation":{"chicago":"Kalai, Gil, and Zuzana Patakova. “Intersection Patterns of Planar Sets.” Discrete and Computational Geometry. Springer Nature, 2020. https://doi.org/10.1007/s00454-020-00205-z.","ista":"Kalai G, Patakova Z. 2020. Intersection patterns of planar sets. Discrete and Computational Geometry. 64, 304–323.","mla":"Kalai, Gil, and Zuzana Patakova. “Intersection Patterns of Planar Sets.” Discrete and Computational Geometry, vol. 64, Springer Nature, 2020, pp. 304–23, doi:10.1007/s00454-020-00205-z.","ieee":"G. Kalai and Z. Patakova, “Intersection patterns of planar sets,” Discrete and Computational Geometry, vol. 64. Springer Nature, pp. 304–323, 2020.","short":"G. Kalai, Z. Patakova, Discrete and Computational Geometry 64 (2020) 304–323.","ama":"Kalai G, Patakova Z. Intersection patterns of planar sets. Discrete and Computational Geometry. 2020;64:304-323. doi:10.1007/s00454-020-00205-z","apa":"Kalai, G., & Patakova, Z. (2020). Intersection patterns of planar sets. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00205-z"},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","article_processing_charge":"No","external_id":{"isi":["000537329400001"],"arxiv":["1907.00885"]},"author":[{"last_name":"Kalai","full_name":"Kalai, Gil","first_name":"Gil"},{"first_name":"Zuzana","id":"48B57058-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-3975-1683","full_name":"Patakova, Zuzana","last_name":"Patakova"}],"title":"Intersection patterns of planar sets"},{"type":"journal_article","article_type":"original","status":"public","_id":"7962","department":[{"_id":"HeEd"}],"date_updated":"2023-08-21T08:49:18Z","scopus_import":"1","main_file_link":[{"url":"https://arxiv.org/abs/1803.06710","open_access":"1"}],"month":"06","intvolume":" 63","abstract":[{"lang":"eng","text":"A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair of them is not connected by any edge (n→∞). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that almost all string graphs on n vertices are intersection graphs of plane convex sets."}],"oa_version":"Preprint","volume":63,"issue":"4","publication_identifier":{"eissn":["14320444"],"issn":["01795376"]},"publication_status":"published","language":[{"iso":"eng"}],"project":[{"grant_number":"Z00342","name":"The Wittgenstein Prize","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"author":[{"first_name":"János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","last_name":"Pach","full_name":"Pach, János"},{"first_name":"Bruce","last_name":"Reed","full_name":"Reed, Bruce"},{"full_name":"Yuditsky, Yelena","last_name":"Yuditsky","first_name":"Yelena"}],"article_processing_charge":"No","external_id":{"isi":["000538229000001"],"arxiv":["1803.06710"]},"title":"Almost all string graphs are intersection graphs of plane convex sets","citation":{"ista":"Pach J, Reed B, Yuditsky Y. 2020. Almost all string graphs are intersection graphs of plane convex sets. Discrete and Computational Geometry. 63(4), 888–917.","chicago":"Pach, János, Bruce Reed, and Yelena Yuditsky. “Almost All String Graphs Are Intersection Graphs of Plane Convex Sets.” Discrete and Computational Geometry. Springer Nature, 2020. https://doi.org/10.1007/s00454-020-00213-z.","short":"J. Pach, B. Reed, Y. Yuditsky, Discrete and Computational Geometry 63 (2020) 888–917.","ieee":"J. Pach, B. Reed, and Y. Yuditsky, “Almost all string graphs are intersection graphs of plane convex sets,” Discrete and Computational Geometry, vol. 63, no. 4. Springer Nature, pp. 888–917, 2020.","ama":"Pach J, Reed B, Yuditsky Y. Almost all string graphs are intersection graphs of plane convex sets. Discrete and Computational Geometry. 2020;63(4):888-917. doi:10.1007/s00454-020-00213-z","apa":"Pach, J., Reed, B., & Yuditsky, Y. (2020). Almost all string graphs are intersection graphs of plane convex sets. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00213-z","mla":"Pach, János, et al. “Almost All String Graphs Are Intersection Graphs of Plane Convex Sets.” Discrete and Computational Geometry, vol. 63, no. 4, Springer Nature, 2020, pp. 888–917, doi:10.1007/s00454-020-00213-z."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","publisher":"Springer Nature","quality_controlled":"1","oa":1,"page":"888-917","date_published":"2020-06-05T00:00:00Z","doi":"10.1007/s00454-020-00213-z","date_created":"2020-06-14T22:00:51Z","isi":1,"year":"2020","day":"05","publication":"Discrete and Computational Geometry"},{"year":"2020","isi":1,"publication":"Discrete and Computational Geometry","day":"01","page":"571-574","date_created":"2020-08-30T22:01:12Z","doi":"10.1007/s00454-020-00237-5","date_published":"2020-10-01T00:00:00Z","oa":1,"publisher":"Springer Nature","citation":{"mla":"Pach, János. “A Farewell to Ricky Pollack.” Discrete and Computational Geometry, vol. 64, Springer Nature, 2020, pp. 571–74, doi:10.1007/s00454-020-00237-5.","short":"J. Pach, Discrete and Computational Geometry 64 (2020) 571–574.","ieee":"J. Pach, “A farewell to Ricky Pollack,” Discrete and Computational Geometry, vol. 64. Springer Nature, pp. 571–574, 2020.","ama":"Pach J. A farewell to Ricky Pollack. Discrete and Computational Geometry. 2020;64:571-574. doi:10.1007/s00454-020-00237-5","apa":"Pach, J. (2020). A farewell to Ricky Pollack. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00237-5","chicago":"Pach, János. “A Farewell to Ricky Pollack.” Discrete and Computational Geometry. Springer Nature, 2020. https://doi.org/10.1007/s00454-020-00237-5.","ista":"Pach J. 2020. A farewell to Ricky Pollack. Discrete and Computational Geometry. 64, 571–574."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","article_processing_charge":"No","external_id":{"isi":["000561483500001"]},"author":[{"full_name":"Pach, János","last_name":"Pach","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","first_name":"János"}],"title":"A farewell to Ricky Pollack","publication_status":"published","publication_identifier":{"eissn":["14320444"],"issn":["01795376"]},"language":[{"iso":"eng"}],"volume":64,"oa_version":"None","main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s00454-020-00237-5"}],"scopus_import":"1","intvolume":" 64","month":"10","date_updated":"2023-08-22T09:05:04Z","department":[{"_id":"HeEd"}],"_id":"8323","article_type":"letter_note","type":"journal_article","status":"public"},{"issue":"4","volume":62,"related_material":{"record":[{"relation":"dissertation_contains","id":"6287","status":"public"}]},"ec_funded":1,"publication_identifier":{"issn":["01795376"],"eissn":["14320444"]},"publication_status":"published","file":[{"date_created":"2019-02-06T10:10:46Z","file_name":"2018_DiscreteCompGeometry_Edelsbrunner.pdf","creator":"dernst","date_updated":"2020-07-14T12:47:10Z","file_size":599339,"checksum":"f9d00e166efaccb5a76bbcbb4dcea3b4","file_id":"5932","access_level":"open_access","relation":"main_file","content_type":"application/pdf"}],"language":[{"iso":"eng"}],"scopus_import":"1","month":"12","intvolume":" 62","abstract":[{"text":"The order-k Voronoi tessellation of a locally finite set 𝑋⊆ℝ𝑛 decomposes ℝ𝑛 into convex domains whose points have the same k nearest neighbors in X. Assuming X is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a relaxed version of discrete Morse theory and generalize by counting only faces, for which the k nearest points in X are within a given distance threshold.","lang":"eng"}],"oa_version":"Published Version","file_date_updated":"2020-07-14T12:47:10Z","department":[{"_id":"HeEd"}],"date_updated":"2023-09-07T12:07:12Z","ddc":["516"],"type":"journal_article","article_type":"original","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"status":"public","_id":"5678","page":"865–878","date_published":"2019-12-01T00:00:00Z","doi":"10.1007/s00454-018-0049-2","date_created":"2018-12-16T22:59:20Z","isi":1,"has_accepted_license":"1","year":"2019","day":"01","publication":"Discrete and Computational Geometry","quality_controlled":"1","publisher":"Springer","oa":1,"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"orcid":"0000-0002-0659-3201","full_name":"Nikitenko, Anton","last_name":"Nikitenko","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","first_name":"Anton"}],"article_processing_charge":"Yes (via OA deal)","external_id":{"arxiv":["1709.09380"],"isi":["000494042900008"]},"title":"Poisson–Delaunay Mosaics of Order k","citation":{"mla":"Edelsbrunner, Herbert, and Anton Nikitenko. “Poisson–Delaunay Mosaics of Order K.” Discrete and Computational Geometry, vol. 62, no. 4, Springer, 2019, pp. 865–878, doi:10.1007/s00454-018-0049-2.","ama":"Edelsbrunner H, Nikitenko A. Poisson–Delaunay Mosaics of Order k. Discrete and Computational Geometry. 2019;62(4):865–878. doi:10.1007/s00454-018-0049-2","apa":"Edelsbrunner, H., & Nikitenko, A. (2019). Poisson–Delaunay Mosaics of Order k. Discrete and Computational Geometry. Springer. https://doi.org/10.1007/s00454-018-0049-2","short":"H. Edelsbrunner, A. Nikitenko, Discrete and Computational Geometry 62 (2019) 865–878.","ieee":"H. Edelsbrunner and A. Nikitenko, “Poisson–Delaunay Mosaics of Order k,” Discrete and Computational Geometry, vol. 62, no. 4. Springer, pp. 865–878, 2019.","chicago":"Edelsbrunner, Herbert, and Anton Nikitenko. “Poisson–Delaunay Mosaics of Order K.” Discrete and Computational Geometry. Springer, 2019. https://doi.org/10.1007/s00454-018-0049-2.","ista":"Edelsbrunner H, Nikitenko A. 2019. Poisson–Delaunay Mosaics of Order k. Discrete and Computational Geometry. 62(4), 865–878."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"},{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}]}]