The Euclidean distance completion problem: cycle completability

The Euclidean distance matrix completion problem asks when a partial distance matrix has a distance matrix completion, in the event that the graph of the specified data is chordal no additional information is needed. If the graph is not chordal, more must be known about the data. In the event the data comprises a full cycle, the additional conditions are quite simple. We characterize those graphs such that the cycle conditions on all minimal cycles imply that a partial distance matrix has a distance matrix completion. One description of these graphs is that they have chordal supergraphs in which no 4-clique includes an added edge, the same condition that appeared in the corresponding question about positive definite completions