On Planar Straight-Line Dominance Drawings

TL;DR

This paper investigates the existence and construction methods of planar straight-line dominance drawings for st-planar graphs, revealing limitations and identifying specific graph classes that always admit such drawings.

Contribution

It demonstrates the non-existence of certain dominance drawings with fixed y-coordinates and introduces classes of st-planar graphs that always admit these drawings.

Findings

Existence of non-constructible dominance drawings with prescribed y-coordinates.

Identification of graph classes always admitting dominance drawings.

Limitations of inductive construction methods for dominance drawings.

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.