Who Needs Crossings?: Noncrossing Linkages are Universal, and Deciding (Global) Rigidity is Hard

2025-10-20

AI Summary

This paper systematically investigates the computational complexity of realizability, rigidity, and global rigidity for three classes of unit-distance graphs: globally non-crossing graphs, matchstick graphs, and general (crossing-permitted) unit-distance graphs. Using the existential theory of the reals (∃ℝ) and universal theory of the reals (∀ℝ), combined with geometric constraint system modeling and algebraic curve tracing techniques, the authors establish foundational results. First, they prove that non-crossing linkages can exactly trace arbitrary polynomial curves—establishing a non-crossing analogue of Kempe’s universality theorem and resolving an open problem dating back to 2004. Second, they show such linkages can generate all compact semialgebraic sets. The work yields nine tight complexity characterizations, eight of which are novel; all are ∃ℝ- or ∀ℝ-complete. These results demonstrate that imposing global non-crossing constraints does not diminish expressive power or drawing capability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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
• Security and Privacy: Large-scale security measurements

Abstract

We exactly settle the complexity of graph realization, graph rigidity, and graph global rigidity as applied to three types of graphs: "globally noncrossing" graphs, which avoid crossings in all of their configurations; matchstick graphs, with unit-length edges and where only noncrossing configurations are considered; and unrestricted graphs (crossings allowed) with unit edge lengths (or in the global rigidity case, edge lengths in ${1,2}$). We show that all nine of these questions are complete for the class $existsmathbb{R}$, defined by the Existential Theory of the Reals, or its complement $forallmathbb{R}$; in particular, each problem is (co)NP-hard. One of these nine results--that realization of unit-distance graphs is $existsmathbb{R}$-complete--was shown previously by Schaefer (2013), but the other eight are new. We strengthen several prior results. Matchstick graph realization was known to be NP-hard (Eades & Wormald 1990, or Cabello et al. 2007), but its membership in NP remained open; we show it is complete for the (possibly) larger class $existsmathbb{R}$. Global rigidity of graphs with edge lengths in ${1,2}$ was known to be coNP-hard (Saxe 1979); we show it is $forallmathbb{R}$-complete. The majority of the paper is devoted to proving an analog of Kempe's Universality Theorem--informally, "there is a linkage to sign your name"--for globally noncrossing linkages. In particular, we show that any polynomial curve $φ(x,y)=0$ can be traced by a noncrossing linkage, settling an open problem from 2004. More generally, we show that the regions in the plane that may be traced by a noncrossing linkage are precisely the compact semialgebraic regions (plus the trivial case of the entire plane). Thus, no drawing power is lost by restricting to noncrossing linkages. We prove analogous results for matchstick linkages and unit-distance linkages as well.

Problem

• Determining complexity of graph realization and rigidity problems
• Proving noncrossing linkages can trace any polynomial curve
• Establishing completeness results for existential theory of reals

Innovation

• Noncrossing linkages trace polynomial curves universally
• Matchstick graph realization is complete for Existential Theory of the Reals
• Global rigidity with edge lengths in {1,2} is complete for Universal Theory of the Reals