Partitioning a Graph into Disjoint Cliques and a Triangle-free Graph

Faisal N. Abu-Khzam; Carl Feghali; Haiko Müller

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arXiv · 1403.5961

Partitioning a Graph into Disjoint Cliques and a Triangle-free Graph

Faisal N. Abu-Khzam·Carl Feghali·Haiko Müller

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Abstract

A graph $G = (V, E)$ is \emph{partitionable} if there exists a partition $\{A, B\}$ of $V$ such that $A$ induces a disjoint union of cliques and $B$ induces a triangle-free graph. In this paper we investigate the computational complexity of deciding whether a graph is partitionable. The problem is known to be $\NP$-complete on arbitrary graphs. Here it is proved that if a graph $G$ is bull-free, planar, perfect, $K_4$-free or does not contain certain holes then deciding whether $G$ is partitionable is $\NP$-complete. This answers an open question posed by Thomass{\'e}, Trotignon and Vu\v{s}kovi{\'c}. In contrast a finite list of forbidden induced subgraphs is given for partitionable cographs.

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Faisal N. Abu-Khzam, Carl Feghali, Haiko Müller. 2014-03-24. Partitioning a Graph into Disjoint Cliques and a Triangle-free Graph. https://arxiv.org/abs/1403.5961

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