# 2-subcoloring is NP-complete for planar comparability graphs

## TL;DR

This paper proves that the problem of 2-subcoloring remains NP-complete even when restricted to planar comparability graphs with maximum degree 4, extending the known complexity results to this class.

## Contribution

It establishes NP-completeness of 2-subcoloring for planar comparability graphs with maximum degree 4, a previously unknown complexity result for this class.

## Key findings

- 2-subcoloring is NP-complete for planar comparability graphs with maximum degree 4.
- Extends NP-completeness results to a new subclass of graphs.
- Shows the problem remains hard under planar and degree constraints.

## Abstract

A $k$-subcoloring of a graph is a partition of the vertex set into at most $k$ cluster graphs, that is, graphs with no induced $P_3$. 2-subcoloring is known to be NP-complete for comparability graphs and three subclasses of planar graphs, namely triangle-free planar graphs with maximum degree 4, planar perfect graphs with maximum degree 4, and planar graphs with girth 5. We show that 2-subcoloring is also NP-complete for planar comparability graphs with maximum degree 4.