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On Planar Straight-Line Dominance Drawings

Patrizio Angelini, Michael A. Bekos, Giuseppe Di Battista, Fabrizio Frati, Luca Grilli, Giacomo Ortali

TL;DR

The paper investigates whether every $st$-planar graph admits a planar straight-line dominance drawing, showing two fundamental barriers: (i) prescribed $y$-coordinates can preclude such drawings, and (ii) the traditional contract-draw-expand method cannot always produce dominance drawings. It provides both negative results for general construction strategies and positive results for several structured graph classes, notably various $st$-plane $3$-tree subclasses, left-non-transitive graphs, and span-2 level graphs. The work introduces techniques including ear decompositions and backward/augmentation constructions to obtain dominance drawings while preserving planarity. Overall, it delineates the boundary between intractable general cases and tractable structured families, offering direction for future methods and open problems.

Abstract

We study the following question, which has been considered since the 90's: Does every $st$-planar graph admit a planar straight-line dominance drawing? We show concrete evidence for the difficulty of this question, by proving that, unlike upward planar straight-line drawings, planar straight-line dominance drawings with prescribed $y$-coordinates do not always exist and planar straight-line dominance drawings cannot always be constructed via a contract-draw-expand inductive approach. We also show several classes of $st$-planar graphs that always admit a planar straight-line dominance drawing. These include $st$-planar $3$-trees in which every stacking operation introduces two edges incoming into the new vertex, $st$-planar graphs in which every vertex is adjacent to the sink, $st$-planar graphs in which no face has the left boundary that is a single edge, and $st$-planar graphs that have a leveling with span at most two.

On Planar Straight-Line Dominance Drawings

TL;DR

The paper investigates whether every -planar graph admits a planar straight-line dominance drawing, showing two fundamental barriers: (i) prescribed -coordinates can preclude such drawings, and (ii) the traditional contract-draw-expand method cannot always produce dominance drawings. It provides both negative results for general construction strategies and positive results for several structured graph classes, notably various -plane -tree subclasses, left-non-transitive graphs, and span-2 level graphs. The work introduces techniques including ear decompositions and backward/augmentation constructions to obtain dominance drawings while preserving planarity. Overall, it delineates the boundary between intractable general cases and tractable structured families, offering direction for future methods and open problems.

Abstract

We study the following question, which has been considered since the 90's: Does every -planar graph admit a planar straight-line dominance drawing? We show concrete evidence for the difficulty of this question, by proving that, unlike upward planar straight-line drawings, planar straight-line dominance drawings with prescribed -coordinates do not always exist and planar straight-line dominance drawings cannot always be constructed via a contract-draw-expand inductive approach. We also show several classes of -planar graphs that always admit a planar straight-line dominance drawing. These include -planar -trees in which every stacking operation introduces two edges incoming into the new vertex, -planar graphs in which every vertex is adjacent to the sink, -planar graphs in which no face has the left boundary that is a single edge, and -planar graphs that have a leveling with span at most two.

Paper Structure

This paper contains 12 sections, 8 theorems, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Planar Straight-line Dominance Drawings are Difficult to Get
  4. Constructing Drawings via Contractions and Expansions
  5. Constructing Drawings by Prescribing the $y$-Coordinates
  6. $st$-plane $3$-trees
  7. Upper $st$-plane $3$-trees
  8. Deep $st$-plane $3$-trees
  9. Sink-dominant $st$-plane $3$-trees
  10. Left-non-transitive $st$-plane Graphs
  11. $st$-plane Span-$2$ Graphs
  12. Conclusions and Open Problems

Key Result

Lemma 1

If a directed graph admits a planar straight-line dominance drawing, it also admits a planar straight-line dominance drawing in which no two vertices share the same $x$- or $y$-coordinate.

Figures (3)

  • Figure 1: Four planar straight-line drawings of an $st$-planar graph $G$. (a) A non-upward drawing. (b) An upward drawing. (c) An $xy$-monotone drawing. (d) A dominance drawing.
  • Figure 2: (a) The contraction of an edge $(u,v)$ in a maximal $st$-plane graph. (b) A maximal $st$-plane graph with no dominance-expandable edge. Thin edges are not contractible, while fat edges are contractible but not dominance-expandable; for example, $(1,3)$ is not dominance-expandable, because vertex $2$ is a predecessor of vertex $3$ but not a predecessor of vertex $1$.
  • Figure 3: (a) The graph for the proof of Theorem \ref{['th:fixed-y-counter']}. (b) The rays $\ell_{1,2}$ and $\ell_{3,4}$ diverge.

Theorems & Definitions (8)

  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8