The Planar Slope Number of Planar Partial 3-Trees of Bounded Degree

TL;DR

This paper establishes an upper bound of O(Δ^5) on the number of slopes needed for straight-line planar drawings of partial 3-trees with maximum degree Δ, addressing a question about outerplanar graphs.

Contribution

It proves that planar partial 3-trees can be drawn with at most O(Δ^5) slopes, providing a bound that was previously unknown.

Findings

Planar partial 3-trees have a planar slope number of at most O(Δ^5).

Answers an open question about the slope number for plane maximal outerplanar graphs.

Provides a bound relevant for graph drawing and visualization applications.

Abstract

It is known that every planar graph has a planar embedding where edges are represented by non-crossing straight-line segments. We study the planar slope number, i.e., the minimum number of distinct edge-slopes in such a drawing of a planar graph with maximum degree $\Delta$. We show that the planar slope number of every planar partial 3-tree and also every plane partial 3-tree is at most $O(\Delta^5)$. In particular, we answer the question of Dujmovi\'c et al. [Computational Geometry 38 (3), pp. 194--212 (2007)] whether there is a function $f$ such that plane maximal outerplanar graphs can be drawn using at most $f(\Delta)$ slopes.