Papers › Generalizing Roberts' characterization of unit interval graphs
Generalizing Roberts' characterization of unit interval graphs
Virginia Ardévol Martínez, Romeo Rizzi, Abdallah Saffidine, Florian Sikora, Stéphane Vialette
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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 \emph{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.
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