  We study a fundamental question from graph drawing: given a pair $(G,C)$ of a
graph $G$ and a cycle $C$ in $G$ together with a simple polygon $P$, is there a
straight-line drawing of $G$ inside $P$ which maps $C$ to $P$? We say that such
a drawing of $(G,C)$ respects $P$. We fully characterize those instances
$(G,C)$ which are polygon-universal, that is, they have a drawing that respects
$P$ for any simple (not necessarily convex) polygon $P$. Specifically, we
identify two necessary conditions for an instance to be polygon-universal. Both
conditions are based purely on graph and cycle distances and are easy to check.
We show that these two conditions are also sufficient. Furthermore, if an
instance $(G,C)$ is planar, that is, if there exists a planar drawing of $G$
with $C$ on the outer face, we show that the same conditions guarantee for
every simple polygon $P$ the existence of a planar drawing of $(G,C)$ that
respects $P$. If $(G,C)$ is polygon-universal, then our proofs directly imply a
linear-time algorithm to construct a drawing that respects a given polygon $P$.
