Let $P$ be a polygon in the plane. An "efficient" triangulation of $P$ is one that introduces no new vertices. We require that all introduced edges be straight and inside $P$. Every polygon in the plane has at least one efficient triangulation.
Let $R$ be a regular polygon with $n$ sides and think of it as the archetypical polygon with $n$ sides. Let $F$ be a family of efficient triangulations of $R$. We call $F$ "realizable" if there is a polygon $P$ with $n$ sides and an isomorphism of $R$ to $P$ as polygons so that the family $F$ is carried exactly to the set of efficient triangulations of $P$.
Is there a reasonable characterization of realizable families?