  We explore several problems related to ruled polygons. Given a ruling of a
polygon $P$, we consider the Reeb graph of $P$ induced by the ruling. We define
the Reeb complexity of $P$, which roughly equates to the minimum number of
points necessary to support $P$. We give asymptotically tight bounds on the
Reeb complexity that are also tight up to a small additive constant. When
restricted to the set of parallel rulings, we show that the Reeb complexity can
be computed in polynomial time.
