Graph Drawings with Few Slopes

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 $\Delta \ge 5$ and all large $n$, there is a $\Delta$-regular $n$-vertex graph with slope-number at least $n^{1-\frac{8+\epsilon }{\Delta +4}}$. 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 $O(\log n)$. 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 (http://arxiv.org/abs/math/0606450), planar drawings of graphs with few slopes are also considered.

This paper is submitted to a journal. A preliminary version appeared as "Really Straight Graph Drawings" in the Graph Drawing 2004 conference. Also see our companion paper (http://arxiv.org/abs/math/0606450)

References in corpus (1)

• Drawings of Planar Graphs with Few Slopes and Segments

Cited by in corpus (5)

• Drawings of Planar Graphs with Few Slopes and Segments
• On Tree-Partition-Width
• Decomposition of Geometric Set Systems and Graphs
• Distinct Distances in Graph Drawings
• Bandwidth, expansion, treewidth, separators, and universality for bounded degree graphs