Maximum Cut on Interval Graphs of Interval Count Two is NP-complete

TL;DR

This paper proves that the Maximum Cut problem remains NP-complete on interval graphs with interval count two, extending the known complexity results from unit and higher interval counts.

Contribution

It establishes NP-completeness of Maximum Cut on interval graphs with interval count two, filling a gap between unit interval graphs and those with higher interval counts.

Findings

Maximum Cut is NP-complete on interval graphs of interval count two.

The complexity status remains open for interval count one.

Extends NP-completeness results to a broader class of interval graphs.

Abstract

An interval graph has interval count $\ell$ if it has an interval model, where among every $\ell+1$ intervals there are two that have the same length. Maximum Cut on interval graphs has been found to be NP-complete recently by Adhikary et al. while deciding its complexity on unit interval graphs (graphs with interval count one) remains a longstanding open problem. More recently, de Figueiredo et al. have made an advancement by showing that the problem remains NP-complete on interval graphs of interval count four. In this paper, we show that Maximum Cut is NP-complete even on interval graphs of interval count two.