Maximum cut on interval graphs of interval count four is NP-complete

TL;DR

This paper proves that the MaxCut problem remains NP-complete even on interval graphs with a bounded interval count of four, resolving a long-standing open question in computational complexity.

Contribution

It provides the first NP-completeness proof for MaxCut on interval graphs with interval count four, extending the known complexity results.

Findings

MaxCut is NP-complete on interval graphs with interval count 4.

Previous proofs of polynomiality for lower interval counts were flawed.

The classification of MaxCut complexity on interval graphs with interval count 1 remains open.

Abstract

The computational complexity of the MaxCut problem restricted to interval graphs has been open since the 80's, being one of the problems proposed by Johnson on his Ongoing Guide to NP-completeness, and has been settled as NP-complete only recently by Adhikary, Bose, Mukherjee and Roy. On the other hand, many flawed proofs of polynomiality for MaxCut on the more restrictive class of unit/proper interval graphs (or graphs with interval count 1) have been presented along the years, and the classification of the problem is still unknown. In this paper, we present the first NP-completeness proof for MaxCut when restricted to interval graphs with bounded interval count, namely graphs with interval count 4.