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

TL;DR

This work settles the complexity of Maximum Cut on interval graphs with interval count $2$ by a polynomial reduction from Maximum Cut on cubic graphs. The authors craft a gadget-based reduction using 3-blocks, vertex/edge/join/switch gadgets, and a Zone/Buffer arrangement to enforce a correspondence between maximum cuts in the source and target graphs while keeping the interval count at $2$. Key contributions include detailed gadget stability lemmas and a complete reduction that preserves cut structures, advancing the quest toward unit interval graphs. The results open avenues for further exploration of Maximum Cut in restricted interval-length settings and for fixed-length sets $C$, with potential implications for understanding the boundary between tractable and intractable cases in geometric graph classes.

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.

Figures (10)

• Figure 1: A 3-block and five long intervals.
• Figure 2: Link chains between consecutive vertex gadgets. Squares represent families of intervals with the same endpoints.
• Figure 3: The composition of $H$.
• Figure 4: A cubic graph $G$.
• Figure 5: The graph $H$ of interval count two whose $\textrm{Maximum Cut}$ partitions correspond to $\textrm{Maximum Cut}$ partitions of the cubic graph $G$ from Figure \ref{['fig:cubic graph']}.

Theorems & Definitions (14)

• Theorem 1
• Example 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof