Multiple interval graphs are a well-known generalization of interval graphs introduced in the 1970s to deal with situations arising naturally in scheduling and allocation. A d -interval is the union of d disjoint intervals on the real line, and a graph is a d -interval graph if it is the intersection graph of d -intervals. In particular, it is a unit d -interval graph if it admits a d -interval representation where every interval has unit length. Whereas it has been known for a long time that recognizing 2-interval graphs and other related classes such as 2-track interval graphs is NP -complete, the complexity of recognizing unit 2-interval graphs remains open. Here, we settle this question by proving that the recognition of unit 2-interval graphs is also NP -complete. Our proof technique uses a completely different approach from the other hardness results of recognizing related classes. Furthermore, we extend the result for unit d -interval graphs for any d ≥ 2 , which does not follow directly in graph recognition problems — as an example, it took almost 20 years to close the gap between d = 2 and d > 2 for the recognition of d -track interval graphs. Our result has several implications, including that for every d ≥ 2 , recognizing ( x , … , x ) d -interval graphs and depth r unit d -interval graphs is NP -complete for every x ≥ 11 and every r ≥ 4 .

For any natural number d, a graph G is a (disjoint) d-interval graph if it is the intersection graph of (disjoint) d-intervals, the union of d (disjoint) intervals on the real line. Two important subclasses of d-interval graphs are unit and balanced d-interval graphs (where every interval has unit length or all the intervals associated to a same vertex have the same length, respectively). A celebrated result by Roberts gives a simple characterization of unit interval graphs being exactly claw-free interval graphs. Here, we study the generalization of this characterization for d-interval graphs. In particular, we prove that for any d ⩾ 2, if G is a K_{1,2d+1}-free interval graph, then G is a unit d-interval graph. However, somehow surprisingly, under the same assumptions, G is not always a disjoint unit d-interval graph. This implies that the class of disjoint unit d-interval graphs is strictly included in the class of unit d-interval graphs. Finally, we study the relationships between the classes obtained under disjoint and non-disjoint d-intervals in the balanced case and show that the classes of disjoint balanced 2-intervals and balanced 2-intervals coincide, but this is no longer true for d > 2.

Multiple interval graphs are a well-known generalization of interval graphs introduced in the 1970s to deal with situations arising naturally in scheduling and allocation. A $d$-interval is the union of $d$ intervals on the real line, and a graph is a $d$-interval graph if it is the intersection graph of $d$-intervals. In particular, it is a unit $d$-interval graph if it admits a $d$-interval representation where every interval has unit length. Whereas it has been known for a long time that recognizing 2-interval graphs and other related classes such as 2-track interval graphs is NP-complete, the complexity of recognizing unit 2-interval graphs remains open. Here, we settle this question by proving that the recognition of unit 2-interval graphs is also NP-complete. Our proof technique uses a completely different approach from the other hardness results of recognizing related classes. Furthermore, we extend the result for unit $d$-interval graphs for any $d\geq 2$, which does not follow directly in graph recognition problems --as an example, it took almost 20 years to close the gap between $d=2$ and $d> 2$ for the recognition of $d$-track interval graphs. Our result has several implications, including that recognizing $(x, \dots, x)$ $d$-interval graphs and depth $r$ unit 2-interval graphs is NP-complete for every $x\geq 11$ and every $r\geq 4$.