Obstacle Numbers of Graphs
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- Alpert, H., Koch, C. & Laison, J.D. Discrete Comput Geom (2010) 44: 223. doi:10.1007/s00454-009-9233-8
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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.