I am given a triangulation $T $ of a set of points $S $ in the plane and a disk $D$ which doesn't contain any triangle. If I now look at the subgraph $G(V,E)$ of $T $ whose vertices are the points of $S$ contained in $D$ and whose edges are the edges of $T $ which are fully contained in $D $ then what can I say about $G$, if $|V|$ is sufficiently large, lets say bigger than 4?
Obviously $G$ is acyclic. Furthermore, a bit informal, each vertex $v$ needs to see some portion of the boundary of $D$ on both sides of any path through $v$.
But are there any other characteristics which have to hold for $G$?