International Symposium on Graph Drawing and Network Visualization

Graph Drawing and Network Visualization pp 360-372 | Cite as

Drawing Graphs Using a Small Number of Obstacles

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)

Abstract

An obstacle representation of a graph G is a set of points in the plane representing the vertices of G, together with a set of polygonal obstacles such that two vertices of G are connected by an edge in G if and only if the line segment between the corresponding points avoids all the obstacles. The obstacle number\({{\mathrm{obs}}}(G)\)ofG is the minimum number of obstacles in an obstacle representation of G.

We provide the first non-trivial general upper bound on the obstacle number of graphs by showing that every n-vertex graph G satisfies \({{\mathrm{obs}}}(G) \le 2n\log {n}\). This refutes a conjecture of Mukkamala, Pach, and Pálvölgyi. For bipartite n-vertex graphs, we improve this bound to \(n-1\). Both bounds apply even when the obstacles are required to be convex. We also prove a lower bound \(2^{\varOmega (hn)}\) on the number of n-vertex graphs with obstacle number at most h for \(h<n\) and an asymptotically matching lower bound \(\varOmega (n^{4/3}M^{2/3})\) for the complexity of a collection of \(M \ge \varOmega (n)\) faces in an arrangement of \(n^2\) line segments with 2n endpoints.

Keywords

Obstacle number Geometric drawing Obstacle representation Arrangement of line segments 

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPraha 1Czech Republic

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