Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Level Planarity Is More Difficult Than We Thought

Simon D. Fink, Matthias Pfretzschner, Ignaz Rutter, Peter Stumpf

TL;DR

Level Planarity asks for a crossing-free drawing of $G=(V,E)$ with a level assignment $\ell:V\to\mathbb{N}$ such that edges are $y$-monotone. This paper presents counterexamples showing that three simple quadratic-time algorithms (Randerath et al., Healy & Kuusik, Harrigan & Healy) can yield false negatives or non-level planar drawings. It links the underlying 2-SAT based characterizations to Hanani-Tutte style criteria, clarifying the limitations of these approaches. The results suggest that a correct, simple embedding algorithm for level graphs remains elusive, prompting a re-evaluation of practical embedding strategies and the need for more robust theoretical foundations.

Abstract

We consider three simple quadratic time algorithms for the problem Level Planarity and give a level-planar instance that they either falsely report as negative or for which they output a drawing that is not level planar.

Level Planarity Is More Difficult Than We Thought

TL;DR

Level Planarity asks for a crossing-free drawing of with a level assignment such that edges are -monotone. This paper presents counterexamples showing that three simple quadratic-time algorithms (Randerath et al., Healy & Kuusik, Harrigan & Healy) can yield false negatives or non-level planar drawings. It links the underlying 2-SAT based characterizations to Hanani-Tutte style criteria, clarifying the limitations of these approaches. The results suggest that a correct, simple embedding algorithm for level graphs remains elusive, prompting a re-evaluation of practical embedding strategies and the need for more robust theoretical foundations.

Abstract

We consider three simple quadratic time algorithms for the problem Level Planarity and give a level-planar instance that they either falsely report as negative or for which they output a drawing that is not level planar.
Paper Structure (3 sections, 2 figures)

This paper contains 3 sections, 2 figures.

Table of Contents

  1. Introduction
  2. Randerath et al.
  3. Healy and Kuusik & Harrigan and Healy

Figures (2)

  • Figure 1: (a) A level-planar graph $G$. (b) The green, blue, and red 2-SAT equivalence classes can be greedily assigned in this order. Subsequently, transitive closure forces $a < b$ as well as $i < g$, but the planarity constraints force $a < b \leftrightarrow f < h \leftrightarrow k < l \leftrightarrow g < i$(c), yielding a contradiction.
  • Figure 2: (a) An initial drawing $\mathcal{L}$ for \ref{['fig:fig-a']}. (b) The corresponding labeled ve-graph. Arrows mark the chose DFS entry points, pairs marked as swapped by Algorithm 1 are shown in gray. (c) The processing order for the vertices of the ve-graph in Algorithm 2.