The complexity of partitioning into disjoint cliques and a triangle-free graph

TL;DR

This paper investigates the computational complexity of partitioning graphs into disjoint cliques and triangle-free graphs, proving NP-completeness for various graph classes and intersections, highlighting the problem's difficulty.

Contribution

It extends previous NP-completeness results to new classes and intersections, deepening understanding of the problem's complexity landscape.

Findings

NP-completeness for intersection of two classes of size four

NP-completeness for graphs with maximum degree 4

NP-completeness for line graphs

Abstract

Motivated by Chudnovsky's structure theorem of bull-free graphs, Abu-Khzam, Feghali, and M\"uller have recently proved that deciding if a graph has a vertex partition into disjoint cliques and a triangle-free graph is NP-complete for five graph classes. The problem is trivial for the intersection of these five classes. We prove that the problem is NP-complete for the intersection of two subsets of size four among the five classes. We also show NP-completeness for other small classes, such as graphs with maximum degree 4 and line graphs.