Z_2-genus of graphs and minimum rank of partial symmetric matrices

The \emph{genus} $\mathrm{g}(G)$ of a graph $G$ is the minimum $g$ such that $G$ has an embedding on the orientable surface $Mg$ of genus $g$. A drawing of a graph on a surface is \emph{independently even} if every pair of nonadjacent edges in the drawing crosses an even number of times. The \emph{$\mathbb{Z}2$-genus} of a graph $G$, denoted by $\mathrm{g}0(G)$, is the minimum $g$ such that $G$ has an independently even drawing on $Mg$. By a result of Battle, Harary, Kodama and Youngs from 1962, the graph genus is additive over 2-connected blocks. In 2013, Schaefer and \v{S}tefankovi\v{c} proved that the $\mathbb{Z}2$-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 $G=G1\cup G2$, $G1$ and $G2$ 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 $|\mathrm{g}0(G)-(\mathrm{g}0(G1)+\mathrm{g}0(G2))|\le k+1$. For complete bipartite graphs $K{m,n}$, with $n\ge m\ge 3$, we prove that $\frac{\mathrm{g}0(K{m,n})}{\mathrm{g}(K{m,n})}=1-O(\frac{1}{n})$. Similar results are proved also for the Euler $\mathbb{Z}2$-genus. We express the $\mathbb{Z}2$-genus of a graph using the minimum rank of partial symmetric matrices over $\mathbb{Z}_2$; a problem that might be of independent interest.