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

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

2010-12-19

arxiv.org/abs/1012.4137v1

Mathematics

Combinatorics

34 pages, 18 figures, old version was at Graph Drawing 2009

Scientific paper

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.

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