Graph drawings with few slopes

  • Authors:

  • Vida Dujmović;Matthew Suderman;David R. Wood

  • Affiliations:

  • Department of Mathematics and Statistics, McGill University, Montréal, Canada;McGill Centre for Bioinformatics, School of Computer Science, McGill University, Montréal, Canada;Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2007

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Abstract

The slope-number of a graph G is the minimum number of distinct edge slopes in a straight-line drawing of G in the plane. We prove that for Δ≥5 and all large n, there is a Δ-regular n-vertex graph with slope-number at least . This is the best known lower bound on the slope-number of a graph with bounded degree. We prove upper and lower bounds on the slope-number of complete bipartite graphs. We prove a general upper bound on the slope-number of an arbitrary graph in terms of its bandwidth. It follows that the slope-number of interval graphs, cocomparability graphs, and AT-free graphs is at most a function of the maximum degree. We prove that graphs of bounded degree and bounded treewidth have slope-number at most . Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.