Given a graph $G$, an {\em 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 {\em 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 $\Omega({n}/{\log n})$.
Deep Dive into Lower bounds on the obstacle number of graphs.
Given a graph $G$, an {\em 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 {\em 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 $\Omega({n}/{\log n})$.
Consider a set P of points in the plane and a set of closed polygonal obstacles whose vertices together with the points in P are in general position, that is, no three of them are on a line. The corresponding visibility graph has P as its vertex set, two points p, q ∈ P being connected by an edge if and only if the segment pq does not meet any of the obstacles. Visibility graphs are extensively studied and used in computational geometry, robot motion planning, computer vision, sensor networks, etc.; see [4], [7], [12], [13], [19].
Alpert, Koch, and Laison [2] introduced an interesting new parameter of graphs, closely related to visibility graphs. Given a graph G, we say that a set of points and a set of polygonal obstacles as above constitute an obstacle representation of G, if the corresponding visibility graph is isomorphic to G. A representation with h obstacles is also called an h-obstacle representation. The smallest number of obstacles in an obstacle representation of G is called the obstacle number of G and is denoted by obs(G). Alpert et al. [2] proved that there exist graphs with arbitrarily large obstacle numbers.
Using tools from extremal graph theory, it was shown in [16] that for any fixed h, the number of graphs with obstacle number at most h is 2 o(n 2 ) . Notice that this immediately implies the existence of graphs with arbitrarily large obstacle numbers.
In the present note, we establish some more precise estimates.
(ii) Moreover, the number of graphs on n (labeled) vertices that admit a representation with a set of obstacles having a total of s sides, is at most 2 O(nlogn+slogs) .
In the above bounds, it makes no difference whether we count labeled or unlabeled graphs, because the number of labeled graphs is at most n! = 2 O(nlogn) times the number of unlabeled ones.
It follows from Theorem 1 (i) that for every n, there exists a graph G on n vertices with obstacle number obs(G) ≥ Ω n/log 2 n .
Indeed, as long as 2 O(hn log 2 n) is smaller than 2 Ω (n 2 ) , the total number of (labeled) graphs with n vertices, we can find at least one graph on n vertices with obstacle number h.
Here we show the following slightly stronger bound.
Theorem 2. For every n, there exists a graph G on n vertices with obstacle number obs(G) ≥ Ω (n/logn) .
This comes close to answering the question in [2] whether there exist graphs with n vertices and obstacle number at least n. However, we have no upper bound on the maximum obstacle number of n-vertex graphs, better than O(n 2 ).
Our next theorem answers another question from [2].
A special instance of the obstacle problem has received a lot of attention, due to its connection to the Szemerédi-Trotter theorem on incidences between points and lines [18], [17], and other classical problems in incidence geometry [15]. This is to decide whether the obstacle number of K n , the empty graph on n vertices, is O(n) if the obstacles must be points. The best known upper bound is n2 O( √ log n) is due to Pach [14]; see also Dumitrescu et al. [5], Matoušek [10], and Aloupis et al. [1].
It is an interesting open problem to decide whether the obstacle number of planar graphs can be bounded from above by a constant. For outerplanar graphs, this has been verified by Fulek, Saeedi, and Sarıöz [6], who proved that every outerplanar graph has obstacle number at most 5.
Theorem i is proved in Section i + 1, 1 ≤ i ≤ 3.
We will prove the theorem by a simple counting method. Before turning to the proof, we introduce some terminology. Given any placement (embedding) of the vertices of G in general position in the plane, a straight-line drawing or, in short, a drawing of G consists of the image of the embedding and the set of open line segments connecting all pairs of points that correspond to the edges of G. If there is no danger of confusion, we make no notational difference between the vertices of G and the corresponding points, and between the pairs uv and the corresponding open segments. The complement of the set of all points that correspond to a vertex or belong to at least one edge of G falls into connected components. These components are called the faces of the drawing. Notice that if G has an obstacle representation with a particular placement of its vertex set, then (1) each obstacle must lie entirely in one face of the drawing, and
(2) each non-edge of G must be blocked by at least one of the obstacles.
We start by proving a result about the convex obstacle number (a special case of Theorem 2), as the arguments are simpler here. Then we tackle Theorem 1 using similar methods.
Following Alpert et al., we define the convex obstacle number obs c (G) of a graph G as the minimal number of obstacles in an obstacle representation of G, in which each obstacle is convex.
The idea is to find a short encoding of the obstacle representations of graphs, and to use this to give an upper bound on the number of graphs with low obstacle number. The proof uses the concept of
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