A Characterization of Visibility Graphs for Pseudo-Polygons

In this paper, we give a characterization of the visibility graphs of pseudo-polygons. We first identify some key combinatorial properties of pseudo-polygons, and we then give a set of five necessary conditions based off our identified properties. We then prove that these necessary conditions are also sufficient via a reduction to a characterization of vertex-edge visibility graphs given by O'Rourke and Streinu

Terrain Visibility Graphs: Persistence Is Not Enough

In this paper, we consider the Visibility Graph Recognition and Reconstruction problems in the context of terrains. Here, we are given a graph $G$ with labeled vertices $v_0, v_1, \ldots, v_{n-1}$ such that the labeling corresponds with a Hamiltonian path $H$. $G$ also may contain other edges. We are interested in determining if there is a terrain $T$ with vertices $p_0, p_1, \ldots, p_{n-1}$ such that $G$ is the visibility graph of $T$ and the boundary of $T$ corresponds with $H$. $G$ is said to be persistent if and only if it satisfies the so-called X-property and Bar-property. It is known that every "pseudo-terrain" has a persistent visibility graph and that every persistent graph is the visibility graph for some pseudo-terrain. The connection is not as clear for (geometric) terrains. It is known that the visibility graph of any terrain $T$ is persistent, but it has been unclear whether every persistent graph $G$ has a terrain $T$ such that $G$ is the visibility graph of $T$. There actually have been several papers that claim this to be the case (although no formal proof has ever been published), and recent works made steps towards building a terrain reconstruction algorithm for any persistent graph. In this paper, we show that there exists a persistent graph $G$ that is not the visibility graph for any terrain $T$. This means persistence is not enough by itself to characterize the visibility graphs of terrains, and implies that pseudo-terrains are not stretchable.Comment: To appear in SoCG 202