Characterization of interval monotone graphs

Characterize the graphs in which every geodesic interval is geodesically convex, thereby resolving the stated unsolved characterization problem for interval monotone graphs.

Background

The paper distinguishes the stress interval from the ordinary geodesic interval. Although stress intervals are shown to be stress-convex, geodesic intervals need not be geodesically convex. Graphs for which all geodesic intervals are geodesically convex are called interval monotone graphs, and the authors note that a complete characterization of this class is unresolved.