Really Straight Drawings II: Non-Planar Graphs

We study straight-line drawings of non-planar graphs with few slopes. Interval graphs, co-comparability graphs and AT-free graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with O(log n) slopes. 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. †School of Computer Science, Carleton University, Ottawa, Ontario, Canada ([email protected]). Supported by NSERC. §School of Computer Science, McGill University, Montréal, Québec, Canada ([email protected]). Supported by NSERC. ¶Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Catalunya, Spain ([email protected]). Supported by the Government of Spain grant MEC SB2003-0270. ∗A preliminary version of this paper was published in the Proceedings of the 12th International Symposium on Graph Drawing (GD ’04), Lecture Notes in Computer Science 3383:122–132, Springer, 2004. Research initiated at the International Workshop on Fixed Parameter Tractability in Geometry and Games, organised by Sue Whitesides; Bellairs Research Institute of McGill University, Holetown, Barbados, February 7–13, 2004.