Discrete & Computational Geometry

, Volume 44, Issue 1, pp 223–244

Obstacle Numbers of Graphs

• Hannah Alpert
• Christina Koch
• Joshua D. Laison
Article

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

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Authors and Affiliations

• Hannah Alpert
• 1
• Christina Koch
• 2
• Joshua D. Laison
• 3
1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA