2018/07/18 by Pavel Klavík, Yota Otachi, Jiří Šejnoha · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Complexity and Algorithms in Graphs #Constraint Satisfaction and Optimization #Interval (graph theory) #Indifference graph #Combinatorics #Interval graph #Mathematics #Parameterized complexity #Pathwidth #Chordal graph #Discrete mathematics #Unit interval #Maximal independent set #Generalization #1-planar graph #Graph #Line graph
paper · doi:10.1007/s00453-018-0481-y
openalex publication_date 2018/07/18 · openalex created_date 2022/08/07 · openalex updated_date 2026/07/29
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 Gajarský 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 W[2]-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 Klavík et al. show that it is polynomial-time solvable for k-nested interval graphs.