  Polygons are a paramount data structure in computational geometry. While the
complexity of many algorithms on simple polygons or polygons with holes depends
on the size of the input polygon, the intrinsic complexity of the problems
these algorithms solve is often related to the reflex vertices of the polygon.
In this paper, we give an easy-to-describe linear-time method to replace an
input polygon $\mathcal{P}$ by a polygon $\mathcal{P}'$ such that (1)
$\mathcal{P}'$ contains $\mathcal{P}$, (2) $\mathcal{P}'$ has its reflex
vertices at the same positions as $\mathcal{P}$, and (3) the number of vertices
of $\mathcal{P}'$ is linear in the number of reflex vertices. Since the
solutions of numerous problems on polygons (including shortest paths, geodesic
hulls, separating point sets, and Voronoi diagrams) are equivalent for both
$\mathcal{P}$ and $\mathcal{P}'$, our algorithm can be used as a preprocessing
step for several algorithms and makes their running time dependent on the
number of reflex vertices rather than on the size of $\mathcal{P}$.
