On the Classes of Interval Graphs of Limited Nesting and Count of Lengths

In 1969, Roberts introduced proper and unit interval graphs and proved that these classes are equal. Natural generalizations of unit interval graphs called $k$-length interval graphs were considered in which the number of different lengths of intervals is limited by $k$. Even after decades of research, no insight into their structure is known and the complexity of recognition is open even for $k=2$. We propose generalizations of proper interval graphs called $k$-nested interval graphs in which there are no chains of $k+1$ intervals nested in each other. It is easy to see that $k$-nested interval graphs are a superclass of $k$-length interval graphs. We give a linear-time recognition algorithm for $k$-nested interval graphs. This algorithm adds a missing piece to Gajarsk\'y et al. [FOCS 2015] to show that testing FO properties on interval graphs is FPT with respect to the nesting $k$ and the length of the formula, while the problem is W2-hard when parameterized just by the length of the formula. We show that a generalization of recognition called partial representation extension is NP-hard for $k$-length interval graphs, even when $k=2$, while Klav\'ik et al. show that it is polynomial-time solvable for $k$-nested interval graphs.