Abstract. The obstacle number of a graph G is the smallest number of polygonal obstacles in the plane with the property that the vertices of G can be represented by distinct points such that two of them see each other if and only if the corresponding vertices are joined by an edge. We list three small graphs that require more than one obstacle. Using extremal graph theoretic tools developed by Prömel, Steger, Bollobás, Thomason, and others, we deduce that for any fixed integer h, the total number of graphs on n vertices with obstacle number at most h is at most 2 o(n 2 ) . This implies that there are bipartite graphs with arbitrarily large obstacle number, which answers a question of Alpert, Koch, and Laison [1].
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.
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