Terrain Visibility Graphs and Cyclic Polytope Triangulations

We prove a bijection between the triangulations of the 3-dimensional cyclic polytope $C(n, 3)$ and persistent graphs with $n-2$ vertices. We show that under this bijection, the Stasheff-Tamari orders naturally translate to subgraph inclusion. Moreover, we describe a connection to the second higher Bruhat order $B(n-2, 2)$. We also give an algorithm to efficiently enumerate all persistent graphs on $n-2$ vertices and thus all triangulations of $C(n, 3)$.