ar X iv : c s . D M / 0 40 51 12 v 1 3 1 M ay 2 00 4 Really Straight Graph Drawings ∗

We study straight-line drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). Drawings of non-planar graphs with few slopes are also considered. For example, 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. School of Computer Science, McGill University, Montréal, Canada ({vida,suderman}@cs.mcgill.ca). School of Computer Science, Carleton University, Ottawa, Canada ([email protected]). Department of Applied Mathematics, Charles University, Prague, Czech Republic. 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, Feb. 7-13, 2004. Research supported by NSERC and COMBSTRU.