Graphs with Large Obstacle Numbers

Abstract

Motivated by questions in computer vision and sensor networks, Alpert et al. [3] introduced the following definitions. Given a graph G, an obstacle representation of G is a set of points in the plane representing the vertices of G, together with a set of connected obstacles such that two vertices of G are joined by an edge if an only if the corresponding points can be connected by a segment which avoids all obstacles. The obstacle number of G is the minimum number of obstacles in an obstacle representation of G. It was shown in [3] that there exist graphs of n vertices with obstacle number at least Ω ( √ log n). We use extremal graph theoretic tools to show that (1) there exist graphs of n vertices with obstacle number at least Ω (n/log 2 n), and (2) the total number of graphs on n vertices with bounded obstacle number is at most 2 o(n 2 ) . Better results are proved if we are allowed to use only convex obstacles or polygonal obstacles with a small number of sides.

Unsolved problems in visibility graphs of points, segments, and polygons

“…Regarding the lower bound on obstacle numbers, Alpet et al [8] have showed that there exists a graph of n vertices with obstacle number O( √ log n), which has been improved to O(n/ log 2 n) by Mukkamala et al [81]. The bound becomes O(n/ log n) if the obstacles are restricted to convex polygons.…”

Lower Bounds on the Obstacle Number of Graphs

“…We are indebted to Deniz Sarıöz for many valuable discussions on the subject. A conference version containing some results from this paper and some from [16] appeared in [11].…”

Obstructing Visibilities with One Obstacle

“…If the previous question is true, given an integer o > 1, what is the smallest number of vertices of a graph with obstacle number o? Mukkamala et al [12] showed the first question is true. For the second question, Alpert et al [1] found a 12-vertex graph that needs two obstacles, namely K * 5,7 , where K * m,n with m ≤ n is the complete bipartite graph minus a matching of size m. They also showed that for any m ≤ n, obs(K * m,n ) ≤ 2.…”