An embedding \(i \mapsto p_i\in \mathbb {R}^d\) of the vertices of a graph G is called universally completable if the following holds: For any other embedding \(i\mapsto q_i~\in \mathbb {R}^{k}\) satisfying \(q_i^{T}q_j = p_i^{T}p_j\) for \(i = j\) and i adjacent to j, there exists an isometry mapping \(q_i\) to \(p_i\) for all \( i\in V(G)\). The notion of universal completability ...