  Being motivated by John Tantalo's Planarity Game, we consider straight line
plane drawings of a planar graph $G$ with edge crossings and wonder how
obfuscated such drawings can be. We define $obf(G)$, the obfuscation complexity
of $G$, to be the maximum number of edge crossings in a drawing of $G$.
Relating $obf(G)$ to the distribution of vertex degrees in $G$, we show an
efficient way of constructing a drawing of $G$ with at least $obf(G)/3$ edge
crossings. We prove bounds $(\delta(G)^2/24-o(1))n^2 < \obf G <3 n^2$ for an
$n$-vertex planar graph $G$ with minimum vertex degree $\delta(G)\ge 2$.
  The shift complexity of $G$, denoted by $shift(G)$, is the minimum number of
vertex shifts sufficient to eliminate all edge crossings in an arbitrarily
obfuscated drawing of $G$ (after shifting a vertex, all incident edges are
supposed to be redrawn correspondingly). If $\delta(G)\ge 3$, then $shift(G)$
is linear in the number of vertices due to the known fact that the matching
number of $G$ is linear. However, in the case $\delta(G)\ge2$ we notice that
$shift(G)$ can be linear even if the matching number is bounded. As for
computational complexity, we show that, given a drawing $D$ of a planar graph,
it is NP-hard to find an optimum sequence of shifts making $D$ crossing-free.
