On the Structure of Graphs with Low Obstacle Number

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 et al. (Discret Comput Geom doi:10.1007/s00454-009-9233-8, 2009).

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Correspondence to János Pach.

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Research supported by NSA grant 47149-0001, NSF grant CCF-08-30272, Swiss National Science Foundation grant 200021-125287/1, and by the Bernoulli Center at EPFL.

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Pach, J., Sarıöz, D. On the Structure of Graphs with Low Obstacle Number. Graphs and Combinatorics 27, 465–473 (2011). https://doi.org/10.1007/s00373-011-1027-0

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Keywords

  • Obstacle number
  • Visibility graph
  • Hereditary graph property
  • Forbidden induced subgraphs
  • Split graphs
  • Enumeration

Mathematics Subject Classification (1991)

  • 05C62
  • 05C75
  • 68R10