English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In 1969, Roberts introduced proper and unit interval graphs and proved thatthese classes are equal. Natural generalizations of unit interval graphs called$k$-length interval graphs were considered in which the number of differentlengths of intervals is limited by $k$. Even after decades of research, noinsight into their structure is known and the complexity of recognition is openeven for $k=2$. We propose generalizations of proper interval graphs called$k$-nested interval graphs in which there are no chains of $k+1$ intervalsnested in each other. It is easy to see that $k$-nested interval graphs are asuperclass 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 showthat testing FO properties on interval graphs is FPT with respect to thenesting $k$ and the length of the formula, while the problem is W2-hard whenparameterized just by the length of the formula. We show that a generalizationof recognition called partial representation extension is NP-hard for$k$-length interval graphs, even when $k=2$, while Klav\'ik et al. show that itis polynomial-time solvable for $k$-nested interval graphs.

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