Characterizing globally linked pairs in graphs

TL;DR

This paper provides a complete combinatorial characterization of globally linked vertex pairs in 2D graphs, confirming longstanding conjectures and offering efficient algorithms, with extensions to higher dimensions and related concepts.

Contribution

It solves a 2006 conjecture by Jackson, Jordan, and Szabadka by characterizing globally linked pairs in 2D graphs and extends results to higher dimensions and related frameworks.

Findings

Complete characterization of globally linked pairs in  graphs.

Confirmation that globally linked pairs, stress-linked pairs, and globally rigid graphs coincide in .

Extension of results to body-bar frameworks in higher dimensions.

Abstract

A pair $\{u,v\}$ of vertices is said to be globally linked in a $d$-dimensional framework $(G,p)$ if there exists no other framework $(G,q)$ with the same edge lengths, in which the distance between the points corresponding to $u$ and $v$ is different from that in $(G,p)$. We say that $\{u,v\}$ is globally linked in $G$ in $\R^d$ if $\{u,v\}$ is globally linked in every generic $d$-dimensional framework $(G,p)$. We give a complete combinatorial characterization of globally linked vertex pairs in graphs in $\R^2$, solving a conjecture of Jackson, Jord\'an and Szabadka from 2006 in the affirmative. Our result provides a refinement of the characterization of globally rigid graphs in $\R^2$ as well as an efficient algorithm for finding the globally linked pairs in a graph. We can also deduce that globally linked pairs in $\R^2$, globally linked pairs in ${\mathbb C}^2$, and stress-linked pairs in ${\mathbb R}^2$ are all the same, settling conjectures of Jackson and Owen, and Garamv\"olgyi, respectively. In higher dimensions we determine the globally linked pairs in body-bar graphs in $\R^d$, for all $d\geq 1$, verifying a conjecture of Connelly, Jord\'an and Whiteley.