A terrain is an x-monotone polygonal line in the xy-plane. Two vertices of a terrain are mutually visible if and only if there is no terrain vertex on or above the open line segment connecting them. A graph whose vertices represent terrain vertices and whose edges represent mutually visible pairs of terrain vertices is called a terrain visibility graph. We would like to find properties that are...

Given a 3-SAT formula, a graph can be constructed in polynomial time such that the graph is a point visibility graph if and only if the 3-SAT formula is satisfiable. This reduction establishes that the problem of recognition of point visibility graphs is NP-hard.

Let s denote a compact convex object in IR. The f-width of s is the perpendicular distance between two distinct parallel lines of support of s with direction f . A set of disjoint convex compact objects in IR is of equal f -width if there exists a direction f such that every pair of objects have equal f -width. A visibility matching, for a set of equal f -width objects is a matching using non-c...

It was shown recently that in the segment endpoint visibility graph Vis(S) of any set S of n disjoint line segments in the plane, there is an alternating path of length (log n), and this bound best possible apart from a constant factor. This talk focuses on the variant of the problem where S is a set of n disjoint axis-parallel line segments, and shows that the length of a longest alternating p...

We consider the problem of routing on a network in the presence of line segment constraints (i.e., obstacles that edges in our network are not allowed to cross). Let P be a set of n points in the plane and let S be a set of non-crossing line segments whose endpoints are in P . We present two deterministic 1-local O(1)-memory routing algorithms that are guaranteed to find a path of at most linea...

This thesis is concerned with simple agents that move from vertex to vertex along straight lines inside a simple polygon with the goal of reconstructing the visibility graph. The visibility graph has a node for each vertex of the polygon with an edge between two nodes if the corresponding vertices see each other, i.e., if they can be connected by a straight line inside the polygon. While at a v...

The paper considers representations of unions and products of trees as rectangle-visibility graphs (abbreviated RVGs), i.e., graphs whose vertices are rectangles in the plane, with adjacency determined by horizontal and vertical visibility. Our main results are that the union of any tree (or forest) with a depth-1 tree is an RVG, and that the union of two depth-2 trees and the union of a depth-...