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.
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机译：图G的G（g）是最小G，使得G在G的可取向表面M_G上嵌入G.即使图纸中的每对非加入边缘交叉偶数次数，也可以独立地独立于曲面图形。由G_0（G）表示的图G的Z_2-GRA，是最小G，使得G具有独立地绘制在M_G上。通过1962年的战斗，Harary，Kodama和Youngs的结果，图表属在2个连接的块上是附加的。 2013年，Schaefer和Stefankovic证明了图表的Z_2-Genus也在2个连接的块上是附加的，并询问该结果是否可以扩展到所谓的2-Amalgamations，作为甲板，格洛弗的结果的类似结果，huneke和stahl for genus。我们给出以下部分答案。如果g = g_1 cup g_2，g_1和g_2在两个顶点u和v中相交，并且guv具有k连接的组件（如果存在的话，我们计算边缘uv），那么| g_0（g） - （g_0（g_0（g_1）+ g_0 （G_2））| <= k + 1。对于完整的二分钟图k_ {m，n}，使用n> = m> = 3，我们证明了g_0（k_ {m，n}）/ g（k_ {m，n}）= 1-o（1 / n ）。对于欧拉Z_2-GRE，还证明了类似的结果。我们使用Z_2上的部分对称矩阵的最小等级来表达图的Z_2-GREN;一个可能是独立兴趣的问题。
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