Polynomial-time recognition of interval graphs that are also disjoint d-interval graphs

Determine whether the recognition problem for the class of graphs that are both interval graphs and disjoint d-interval graphs admits a polynomial-time algorithm for general d ≥ 2.

Background

While interval graphs can be recognized in polynomial time, recognizing d-interval graphs is NP-complete for d ≥ 2 in general. The paper proves that a key characterization fails in the disjoint setting, prompting the question of whether the intersection class (graphs that are both interval and disjoint d-interval) might still be efficiently recognizable.

Resolving this question would clarify algorithmic tractability for a notable subclass motivated by the failure of a straightforward structural generalization.