{"title":"Z_2-Genus of graphs and minimum rank of partial symmetric matrices","oa_version":"Published Version","status":"public","month":"06","type":"conference","conference":{"end_date":"2019-06-21","start_date":"2019-06-18","location":"Portland, OR, United States","name":"SoCG: Symposium on Computational Geometry"},"day":"01","article_number":"39","publication_status":"published","_id":"7401","tmp":{"short":"CC BY (4.0)","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)"},"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","year":"2019","project":[{"_id":"261FA626-B435-11E9-9278-68D0E5697425","grant_number":"M02281","name":"Eliminating intersections in drawings of graphs","call_identifier":"FWF"}],"article_processing_charge":"No","date_updated":"2024-10-09T20:59:14Z","doi":"10.4230/LIPICS.SOCG.2019.39","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","external_id":{"arxiv":["1903.08637"]},"abstract":[{"lang":"eng","text":"The genus g(G) of a graph G is the minimum g such that G has an embedding on the orientable surface M_g of genus g. A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The Z_2-genus of a graph G, denoted by g_0(G), is the minimum g such that G has an independently even drawing on M_g. By a result of Battle, Harary, Kodama and Youngs from 1962, the graph genus is additive over 2-connected blocks. In 2013, Schaefer and Stefankovic proved that the Z_2-genus of a graph is additive over 2-connected blocks as well, and asked whether this result can be extended to so-called 2-amalgamations, as an analogue of results by Decker, Glover, Huneke, and Stahl for the genus. We give the following partial answer. If G=G_1 cup G_2, G_1 and G_2 intersect in two vertices u and v, and G-u-v has k connected components (among which we count the edge uv if present), then |g_0(G)-(g_0(G_1)+g_0(G_2))|<=k+1. For complete bipartite graphs K_{m,n}, with n >= m >= 3, we prove that g_0(K_{m,n})/g(K_{m,n})=1-O(1/n). Similar results are proved also for the Euler Z_2-genus. We express the Z_2-genus of a graph using the minimum rank of partial symmetric matrices over Z_2; a problem that might be of independent interest. "}],"ddc":["000"],"has_accepted_license":"1","quality_controlled":"1","department":[{"_id":"UlWa"}],"scopus_import":1,"file":[{"date_updated":"2020-07-14T12:47:57Z","file_id":"7445","checksum":"aac37b09118cc0ab58cf77129e691f8c","date_created":"2020-02-04T09:14:31Z","file_name":"2019_LIPIcs_Fulek.pdf","file_size":628347,"creator":"dernst","content_type":"application/pdf","access_level":"open_access","relation":"main_file"}],"author":[{"first_name":"Radoslav","full_name":"Fulek, Radoslav","last_name":"Fulek","id":"39F3FFE4-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-8485-1774"},{"last_name":"Kyncl","first_name":"Jan","full_name":"Kyncl, Jan"}],"intvolume":"       129","publication":"35th International Symposium on Computational Geometry (SoCG 2019)","language":[{"iso":"eng"}],"corr_author":"1","date_created":"2020-01-29T16:17:05Z","date_published":"2019-06-01T00:00:00Z","alternative_title":["LIPIcs"],"oa":1,"arxiv":1,"citation":{"short":"R. Fulek, J. Kyncl, in:, 35th International Symposium on Computational Geometry (SoCG 2019), Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019.","mla":"Fulek, Radoslav, and Jan Kyncl. “Z_2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices.” <i>35th International Symposium on Computational Geometry (SoCG 2019)</i>, vol. 129, 39, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019, doi:<a href=\"https://doi.org/10.4230/LIPICS.SOCG.2019.39\">10.4230/LIPICS.SOCG.2019.39</a>.","ama":"Fulek R, Kyncl J. Z_2-Genus of graphs and minimum rank of partial symmetric matrices. In: <i>35th International Symposium on Computational Geometry (SoCG 2019)</i>. Vol 129. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2019. doi:<a href=\"https://doi.org/10.4230/LIPICS.SOCG.2019.39\">10.4230/LIPICS.SOCG.2019.39</a>","apa":"Fulek, R., &#38; Kyncl, J. (2019). Z_2-Genus of graphs and minimum rank of partial symmetric matrices. In <i>35th International Symposium on Computational Geometry (SoCG 2019)</i> (Vol. 129). Portland, OR, United States: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPICS.SOCG.2019.39\">https://doi.org/10.4230/LIPICS.SOCG.2019.39</a>","chicago":"Fulek, Radoslav, and Jan Kyncl. “Z_2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices.” In <i>35th International Symposium on Computational Geometry (SoCG 2019)</i>, Vol. 129. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. <a href=\"https://doi.org/10.4230/LIPICS.SOCG.2019.39\">https://doi.org/10.4230/LIPICS.SOCG.2019.39</a>.","ista":"Fulek R, Kyncl J. 2019. Z_2-Genus of graphs and minimum rank of partial symmetric matrices. 35th International Symposium on Computational Geometry (SoCG 2019). SoCG: Symposium on Computational Geometry, LIPIcs, vol. 129, 39.","ieee":"R. Fulek and J. Kyncl, “Z_2-Genus of graphs and minimum rank of partial symmetric matrices,” in <i>35th International Symposium on Computational Geometry (SoCG 2019)</i>, Portland, OR, United States, 2019, vol. 129."},"file_date_updated":"2020-07-14T12:47:57Z","volume":129,"publication_identifier":{"issn":["1868-8969"],"isbn":["978-3-95977-104-7"]}}