Graphs with Large Obstacle Numbers

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6410)


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 \(\Omega(\sqrt{\log n})\). We use extremal graph theoretic tools to show that (1) there exist graphs of n vertices with obstacle number at least Ω(n/log2 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.


Computational Geometry Colorable Graph Order Type Visibility Graph Complete Subgraph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.RutgersThe State University of New JerseyPiscatawayUSA
  2. 2.Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland
  3. 3.The Graduate Center of the City University of New YorkNew YorkUSA

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