Lower Bounds on the Obstacle Number of Graphs

Abstract

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 and 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 is shown that there are graphs on n vertices with obstacle number at least Ω (n/log n).

Graphs with Obstacle Number Greater than One

“…The study of the obstacle number per se was initiated by Alpert, Koch, & Laison [2]. These parameters have since been investigated by others [8,10,16,17,18,19]. Alpert, Koch, & Laison [2,Thm.…”

On Obstacle Numbers

“…Alpert et al initially show that the worst-case obstacle number is Ω( log n/ log log n) and posed as an open problem the question of determining if w(n) ∈ Ω(n). Mukkamala et al [13] showed that w(n) ∈ Ω(n/ log 2 n) and Mukkamala et al [12] later increased this to w(n) ∈ Ω(n/ log n). In the current paper, we up the lower-bound again by proving the following theorem: Theorem 1.…”

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The obstacle number is a new graph parameter introduced by Alpert, Koch, and Laison (2010). Mukkamala et al (2012) show that there exist graphs with n vertices having obstacle number in Ω(n/ log n). In this note, we up this lower bound to Ω(n/(log log n) 2 ).

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Drawing Graphs Using a Small Number of Obstacles

“…2 ) denotes the complement of G. On the other hand, a modification of the proof of the lower bound by Mukkamala, Pach, and Pálvölgyi [13] implies that there are bipartite graphs G on n vertices with obs(G) ≥ Ω(n/ log n) for every positive integer n.…”

“…Mukkamala, Pach, and Sarıöz [14] established more precise bounds by showing that the number of labeled n-vertex graphs with obstacle number at most h is at most 2 O(hn log 2 n) for every fixed positive integer h. It follows that obs(n) ≥ Ω(n/ log 2 n). Later, Mukkamala, Pach, and Pálvölgyi [13] improved the lower bound to obs(n) ≥ Ω(n/ log n). Currently, the strongest lower bound on the obstacle number is due to Dujmović and Morin [7] who showed obs(n) ≥ Ω(n/(log log n) 2 ).…”