Graphs with single interval Cayley configuration spaces in 3-dimensions

We prove a conjectured graph theoretic characterization of a geometric property of 3 dimensional linkages posed 15 years ago by Sitharam and Gao, motivated by their equivalent characterization for $d\le 2$ that does not generalize to $d\ge 3$. A linkage $(G,\ell)$ contains a finite simple undirected graph $G$ and a map $\ell$ that assigns squared Euclidean lengths to the edges of $G$. A \emph{$d$-realization} of $(G,\ell)$ is an assignment of points in $\mathbb{R}^d$ to the vertices of $G$ for which pairwise squared distances between points agree with $\ell$. For any positive integer $d \leq 3$, we characterize pairs $(G,f)$, where $f$ is a nonedge of $G$, such that, for any linkage $(G,\ell)$, the lengths attained by $f$ form a single interval - over the (typically a disconnected set of) $d$-realizations of $(G,\ell)$. Although related to the minor closed class of $d$-flattenable graphs, the class of pairs $(G,f)$ with the above property is not closed under edge deletions, has no obvious well quasi-ordering, and there are infinitely many minimal graph-nonedge pairs - with respect to edge contractions - in the complement class. Our characterization overcomes these obstacles, is based on the forbidden minors for $d$-flattenability for $d \leq 3$, and contributes to the theory of Cayley configurations with many applications. Helper results and corollaries provide new tools for reasoning about configuration spaces and completions of partial 3-tree linkages, (non)convexity of Euclidean measurement sets in $3$-dimensions, their projections, fibers and sections. Generalizations to higher dimensions and efficient algorithmic characterizations are conjectured.