Characterize interval graphs that are also disjoint d-interval graphs

Characterize, for each integer d ≥ 2, the class of graphs that are both interval graphs and disjoint d-interval graphs by identifying definitive structural or representation-based criteria, in order to extend characterization results to this setting.

Background

The paper generalizes Roberts’ characterization of unit interval graphs to unit d-interval graphs under the general (non-disjoint) definition, but shows that the same characterization fails for disjoint unit d-interval graphs. This reveals a gap in understanding the structural description of graphs that simultaneously lie in both classes: interval graphs and disjoint d-interval graphs.

Because the natural generalization does not hold for disjoint intervals, the authors raise the need for an alternative characterization specific to the intersection of these classes.