  The \emph{genus} $\mathrm{g}(G)$ 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 \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 $M_g$.
  By a result of Battle, Harary, Kodama and Youngs from 1962, the graph genus
is additive over 2-connected blocks.
  In 2013, Schaefer and \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 $G=G_1\cup G_2$, $G_1$ 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
$|\mathrm{g}_0(G)-(\mathrm{g}_0(G_1)+\mathrm{g}_0(G_2))|\le k+1$.
  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.

Combinatorics;Computational Geometry;Discrete Mathematics