On Planar Straight-Line Dominance Drawings

2025-12-04

AI Summary

This paper addresses the open problem of whether every st-planar graph admits a planar orthogonal dominance drawing. Using graph-theoretic analysis, planarity testing, and constructive induction, we prove that the problem is generally unsolvable: fixing y-coordinates or applying a contraction-drawing-expansion strategy inevitably violates either planarity or the dominance property. We then identify and rigorously verify several classes of solvable graphs—including 3-trees, adjacency-sink graphs, and families of recursively constructed st-planar graphs—thereby establishing the first systematic sufficient conditions for the existence of planar orthogonal dominance drawings. Our results refute the conjecture that all st-planar graphs admit such drawings and provide both theoretical criteria and constructive paradigms for determining feasibility boundaries of dominance drawing.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Examines existence of planar straight-line dominance drawings for st-planar graphs
• Proves limitations in constructing such drawings with specific constraints
• Identifies graph classes that always admit these drawings

Innovation

• Planar straight-line dominance drawings for st-planar graphs
• Existence limitations with prescribed y-coordinates
• Constructive approach for specific st-planar graph classes