Discrete & Computational Geometry

, Volume 44, Issue 1, pp 223–244

Obstacle Numbers of Graphs

Article

DOI: 10.1007/s00454-009-9233-8

Cite this article as:
Alpert, H., Koch, C. & Laison, J.D. Discrete Comput Geom (2010) 44: 223. doi:10.1007/s00454-009-9233-8

Abstract

An obstacle representation of a graph G is a drawing of G in the plane with straight-line edges, together with a set of polygons (respectively, convex polygons) called obstacles, such that an edge exists in G if and only if it does not intersect an obstacle. The obstacle number (convex obstacle number) of G is the smallest number of obstacles (convex obstacles) in any obstacle representation of G. In this paper, we identify families of graphs with obstacle number 1 and construct graphs with arbitrarily large obstacle number (convex obstacle number). We prove that a graph has an obstacle representation with a single convex k-gon if and only if it is a circular arc graph with clique covering number at most k in which no two arcs cover the host circle. We also prove independently that a graph has an obstacle representation with a single segment obstacle if and only if it is the complement of an interval bigraph.

Obstacle number Convex obstacle number Circular arc graph Proper circular arc graph Non-double-covering circular arc graph Interval bigraph Visibility graph 

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Hannah Alpert
    • 1
  • Christina Koch
    • 2
  • Joshua D. Laison
    • 3
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Academy of HopeWashingtonUSA
  3. 3.Mathematics DepartmentWillamette UniversitySalemUSA

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