--- res: bibo_abstract: - 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. @eng bibo_authorlist: - foaf_Person: foaf_givenName: Radoslav foaf_name: Fulek, Radoslav foaf_surname: Fulek foaf_workInfoHomepage: http://www.librecat.org/personId=39F3FFE4-F248-11E8-B48F-1D18A9856A87 orcid: 0000-0001-8485-1774 - foaf_Person: foaf_givenName: Jan foaf_name: Kyncl, Jan foaf_surname: Kyncl bibo_doi: 10.4230/LIPICS.SOCG.2019.39 bibo_volume: 129 dct_date: 2019^xs_gYear dct_isPartOf: - http://id.crossref.org/issn/1868-8969 - http://id.crossref.org/issn/978-3-95977-104-7 dct_language: eng dct_publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik@ dct_title: Z_2-Genus of graphs and minimum rank of partial symmetric matrices@ ...