What are some nice ways to think about polygons from a "triangulation" viewpoint ?
For further clarification of what I mean: Define an equivalence relation between two polygons $P_1= A_1A_2 \cdots A_n$ and $P_2= B_1B_2 \cdots B_n$ if there's a permutation $\pi$ such that the segment $\overline{A_iA_j}$ lies completely inside $P_1$ iff the segment $\overline{B_{\pi(i)}B_{\pi(j)}} $ lies completely inside $P_2$. For example, all $n$-sided convex polygon are equivalent. It's also clear two equivalent polygons can be triangulated in the same number of ways.
What's a nice way to think of the elements of the equivalence classes ? One thing I have tried is to create a graph with $\binom{n}{2}$ vertices and an edge between $v_{ij}$ and $v_{kl}$ iff the two edges $\overline{A_iA_j}, \overline{A_kA_l}$ don't interest. But not all graphs correspond to a polygon, and this is kinda clunky. Is there a nicer way to think about them?