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

Abstract: 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.