{"_id":"18556","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_place":"publisher","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."}],"has_accepted_license":"1","year":"2024","oa_version":"Published Version","volume":320,"conference":{"location":"Vienna, Austria","end_date":"2024-09-20","name":"GD: Graph Drawing and Network Visualization","start_date":"2024-09-18"},"ec_funded":1,"arxiv":1,"month":"10","article_number":"3","oa":1,"title":"The Euclidean MST-ratio for bi-colored lattices","publication_identifier":{"isbn":["9783959773430"],"issn":["1868-8969"]},"scopus_import":"1","OA_type":"gold","corr_author":"1","department":[{"_id":"HeEd"}],"file_date_updated":"2024-11-18T07:49:25Z","tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"date_published":"2024-10-28T00:00:00Z","publication":"32nd International Symposium on Graph Drawing and Network Visualization","status":"public","date_created":"2024-11-17T23:01:47Z","file":[{"access_level":"open_access","file_name":"2024_LIPIcs_CultreradiMontesano.pdf","checksum":"5f9b35e115c3d375e99be78da9054cb4","date_updated":"2024-11-18T07:49:25Z","content_type":"application/pdf","file_id":"18560","date_created":"2024-11-18T07:49:25Z","creator":"dernst","file_size":908541,"relation":"main_file","success":1}],"author":[{"id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","full_name":"Cultrera di Montesano, Sebastiano","last_name":"Cultrera di Montesano","orcid":"0000-0001-6249-0832","first_name":"Sebastiano"},{"full_name":"Draganov, Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0464-3823","first_name":"Ondrej","last_name":"Draganov"},{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner"},{"first_name":"Morteza","last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"ddc":["510"],"external_id":{"arxiv":["2403.10204"]},"date_updated":"2024-11-18T07:51:39Z","intvolume":"       320","quality_controlled":"1","day":"28","type":"conference","article_processing_charge":"Yes","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","doi":"10.4230/LIPIcs.GD.2024.3","language":[{"iso":"eng"}],"alternative_title":["LIPIcs"],"citation":{"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>","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.","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>.","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.","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.","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>","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>."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"grant_number":"Z00342","name":"Wittgenstein Award - Herbert Edelsbrunner","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"publication_status":"published"}