Drawing Graphs with Few Arcs

Let G = (V,E) be a planar graph. An arrangement of circular arcs is called a composite arc-drawing of G, if its 1-skeleton is isomorphic to G. Similarly, a composite segment-drawing is described by an arrangement of straight-line segments. We ask for the smallest possible ground set of arcs/segments for a composite arc/segment-drawing. We present algorithms for constructing composite arc-drawings with a small ground set for trees, series-parallel graphs, planar 3-trees and general planar graphs. In the case where G is a tree, we also introduce an algorithm that realizes the vertices of the composite drawing on a O(n)× n grid. For each of the graph classes we provide a lower bound for the maximal size of the arrangement’s ground set. Submitted: December 2013 Reviewed: July 2015 Revised: August 2015 Accepted: August 2015 Final: August 2015 Published: August 2015 Article type: Regular Paper Communicated by: A. Symvonis The research was performed in cooperation with Eurogiga project GraDR Graph Drawings and Representations. It was partially funded by the German Research Foundation DFG under grant SCHU 2458/4-1. A preliminary version has appeared at the 39th InternationalWorkshop on Theoretic Concepts in Computer Science [14]. E-mail address: [email protected] (André Schulz) 394 A. Schulz Drawing Graphs with Few Arcs