Partial Reflections and Globally Linked Pairs in Rigid Graphs

SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2005-2040, September 2024.
Abstract. A [math]-dimensional framework is a pair [math], where [math] is a graph and [math] maps the vertices of [math] to points in [math]. The edges of [math] are mapped to the corresponding line segments. A graph [math] is said to be globally rigid in [math] if every generic [math]-dimensional framework [math] is determined, up to congruence, by its edge lengths. A finer property is global linkedness: we say that a vertex pair [math] of [math] is globally linked in [math] in [math] if in every generic [math]-dimensional framework [math] the distance between [math] and [math] is uniquely determined by the edge lengths. In this paper we investigate globally linked pairs in graphs in [math]. We give several characterizations of those rigid graphs [math] in which a pair [math] is globally linked if and only if there exist [math] internally disjoint paths from [math] to [math] in [math]. We call these graphs [math]-joined. Among others, we show that [math] is [math]-joined if and only if for each pair of generic frameworks of [math] with the same edge lengths, one can be obtained from the other by a sequence of partial reflections along hyperplanes determined by [math]-separators of [math]. We also show that the family of [math]-joined graphs is closed under edge addition, as well as under gluing along [math] or more vertices. As a key ingredient to our main results, we prove that rigid graphs in [math] contain no crossing [math]-separators. Our results give rise to new families of graphs for which global linkedness (and global rigidity) in [math] can be tested in polynomial time.