[Paper Review] 1-bend Upward Planar Drawings of SP-digraphs

Emilio Di Giacomo, Giuseppe Liotta+1 more|arXiv (Cornell University)|Aug 30, 2016

Computational Geometry and Mesh Generation17 references4 citations

TL;DR

This paper presents a linear-time algorithm for constructing 1-bend upward planar drawings of series-parallel digraphs (SP-digraphs) using exactly Δ distinct slopes, where Δ is the maximum vertex degree. The method leverages a contact representation of vertices as crosses and transforms it into a slope-optimized upward drawing, achieving worst-case optimal angular resolution π/Δ and proving Δ slopes are necessary and sufficient for SP-digraphs.

ABSTRACT

It is proved that every series-parallel digraph whose maximum vertex-degree is $Δ$ admits an upward planar drawing with at most one bend per edge such that each edge segment has one of $Δ$ distinct slopes. This is shown to be worst-case optimal in terms of the number of slopes. Furthermore, our construction gives rise to drawings with optimal angular resolution $\fracπΔ$. A variant of the proof technique is used to show that (non-directed) reduced series-parallel graphs and flat series-parallel graphs have a (non-upward) one-bend planar drawing with $\lceil\fracΔ{2} ceil$ distinct slopes if biconnected, and with $\lceil\fracΔ{2} ceil+1$ distinct slopes if connected.

Motivation & Objective

• To determine the minimum number of distinct slopes required for 1-bend upward planar drawings of SP-digraphs.
• To establish tight bounds on the 1-bend upward planar slope number for SP-digraphs and related undirected graph classes.
• To develop a linear-time algorithm that achieves optimal angular resolution and slope usage for upward planar drawings of SP-digraphs.
• To extend the results to undirected reduced and flat SP-graphs, showing that ⌈Δ/2⌉ slopes suffice for biconnected graphs.

Proposed method

• Represent each vertex in the SP-digraph as a cross (intersecting horizontal and vertical segments) in a well-spaced contact representation.
• Assign slopes from the set 𝒮Δ = {π/2 + iπ/Δ | i = 0, ..., Δ−1} to edge segments, ensuring each edge has at most one bend.
• Transform each cross into a polyline drawing by replacing each contact with two segments using slopes from 𝒮Δ, preserving upward direction and planarity.
• Use the safe-region property of each cross to ensure bends occur only within disjoint zones, avoiding edge crossings.
• Handle transitive edges in general SP-digraphs by subdividing them and later removing subdivision vertices to restore original structure.
• Leverage symmetry and slope reuse in undirected graphs to reduce the number of required slopes to ⌈Δ/2⌉ for biconnected reduced or flat SP-graphs.

Experimental results

Research questions

• RQ1 What is the 1-bend upward planar slope number of SP-digraphs with maximum vertex-degree Δ?
• RQ2 Is Δ the worst-case optimal number of slopes for 1-bend upward planar drawings of SP-digraphs?
• RQ3 Can the 1-bend planar slope number of undirected SP-graphs be reduced below the known upper bound of 3/2(Δ−1)?
• RQ4 What is the minimum number of slopes required for 1-bend planar drawings of biconnected reduced or flat SP-graphs?
• RQ5 Can the slope bound be extended to general partial 2-trees beyond SP-graphs?

Key findings

• Every SP-digraph with maximum vertex-degree Δ admits a 1-bend upward planar drawing using exactly Δ distinct slopes, and this bound is worst-case optimal.
• The constructed drawings achieve angular resolution at least π/Δ, which is also worst-case optimal.
• The algorithm runs in O(n) time, making it efficient for large SP-digraphs.
• For biconnected reduced or flat SP-graphs, the 1-bend planar slope number is at most ⌈Δ/2⌉, with angular resolution at least 2π/Δ.
• For connected reduced or flat SP-graphs, the 1-bend planar slope number is at most ⌈Δ/2⌉ + 1, with angular resolution at least 2π/(Δ+1).
• A family of SP-digraphs is constructed to show that Δ slopes are necessary in any 1-bend upward planar drawing, proving tightness of the upper bound.