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

Reconstructing Generalized Staircase Polygons with Uniform Step Length

Visibility graph reconstruction, which asks us to construct a polygon that has a given visibility graph, is a fundamental problem with unknown complexity (although visibility graph recognition is known to be in PSPACE). We show that two classes of uniform step length polygons can be reconstructed efficiently by finding and removing rectangles formed between consecutive convex boundary vertices called tabs. In particular, we give an $O(n^2m)$-time reconstruction algorithm for orthogonally convex polygons, where $n$ and $m$ are the number of vertices and edges in the visibility graph, respectively. We further show that reconstructing a monotone chain of staircases (a histogram) is fixed-parameter tractable, when parameterized on the number of tabs, and polynomially solvable in time $O(n^2m)$ under reasonable alignment restrictions.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Colouring Polygon Visibility Graphs and Their Generalizations

Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ? has chromatic number at most 3?4^{?-1}. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a colouring with the claimed number of colours can be computed in polynomial time

The VC-Dimension of Limited Visibility Terrains

Visibility problems are fundamental to computational geometry, and many versions of geometric set cover where coverage is based on visibility have been considered. In most settings, points can see "infinitely far" so long as visibility is not "blocked" by some obstacle. In many applications, this may be an unreasonable assumption. In this paper, we consider a new model of visibility where no point can see any other point beyond a sight radius ?. In particular, we consider this visibility model in the context of terrains. We show that the VC-dimension of limited visibility terrains is exactly 7. We give lower bound construction that shatters a set of 7 points, and we prove that shattering 8 points is not possible

The identification of archaeological sites by false color infrared aerial photography

The study of color infrared photography of Tehuacan Valley, Mexico was made to determine the applicability of remotely sensed data to archeology. Photography was interpreted without prior knowledge of the area, followed by a field check to determine accuracy of the original interpretations and to evaluate causes of successes and failures. Results indicate that the visibility of sites depends primarily on its environmental situation, and also that the delineation of environments and microenvironments is especially easy with this type of film. Furthermore, the age and size of the sites are not necessarily the deciding factors in their discernment