Polarity and Monopolarity of $3$-colourable comparability graphs

TL;DR

This paper proves that determining polarity and monopolarity remains NP-complete even when restricted to 3-colorable comparability graphs, highlighting the computational difficulty of these problems in a specific graph class.

Contribution

It establishes NP-completeness of polarity and monopolarity for 3-colorable comparability graphs, extending known complexity results to a more restricted graph class.

Findings

Polarity and monopolarity are NP-complete for 3-colorable comparability graphs.

Reduction from 1-3-SAT demonstrates the hardness.

Both problems remain NP-complete in this restricted class.

Abstract

We sharpen the result that polarity and monopolarity are NP-complete problems by showing that they remain NP-complete if the input graph is restricted to be a $3$-colourable comparability graph. We start by presenting a construction reducing $1$-$3$-SAT to monopolarity of $3$-colourable comparability graphs. Then we show that polarity is at least as hard as monopolarity for input graphs restricted to a fixed disjoint-union-closed class. We conclude the paper by stating that both polarity and monopolarity of $3$-colourable comparability graphs are NP-complete problems.