Graphs and Combinatorics

, Volume 27, Issue 3, pp 465–473

On the Structure of Graphs with Low Obstacle Number

Proceedings Paper

DOI: 10.1007/s00373-011-1027-0

Cite this article as:
Pach, J. & Sarıöz, D. Graphs and Combinatorics (2011) 27: 465. doi:10.1007/s00373-011-1027-0

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).

Keywords

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

Mathematics Subject Classification (1991)

05C62 05C75 68R10 

Copyright information

© Springer 2011

Authors and Affiliations

  1. 1.École Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.The Graduate Center of the City University of New YorkNew YorkUSA