  An \emph{obstacle representation} of a graph $G$ is a straight-line drawing
of $G$ in the plane together with a collection of connected subsets of the
plane, called \emph{obstacles}, that block all non-edges of $G$ while not
blocking any of the edges of $G$. The \emph{obstacle number} obs$(G)$ is the
minimum number of obstacles required to represent $G$.
  We study the structure of graphs with obstacle number greater than one. We
show that the icosahedron has obstacle number $2$, thus answering a question of
Alpert, Koch, \& Laison asking whether all planar graphs have obstacle number
at most $1$. We also show that the $1$-skeleton of a related polyhedron, the
\emph{gyroelongated $4$-dipyramid}, has obstacle number $2$. The order of this
graph is $10$, which is also the order of the smallest known graph with
obstacle number $2$.
  Some of our methods involve instances of the Satisfiability problem, we make
use of various "SAT solvers" in order to produce computer-assisted proofs.
