Work

Recognizing unit multiple interval graphs is hard

Year: 2024

Type: article

Abstract: 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 .

Source: Discrete Applied Mathematics

DOI: 10.1016/j.dam.2024.09.011

Authors: Virginia Ardévol Martínez, Romeo Rizzi, Florian Sikora, Stéphane Vialette