Partitioning a Graph into Disjoint Cliques and a Triangle-free Graph
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{é}, Trotignon and Vuškovi{ć}. In contrast a finite list of forbidden induced subgraphs is given for partitionable cographs.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2014
- DOI:
- 10.48550/arXiv.1403.5961
- arXiv:
- arXiv:1403.5961
- Bibcode:
- 2014arXiv1403.5961A
- Keywords:
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- Computer Science - Computational Complexity;
- Computer Science - Discrete Mathematics;
- Mathematics - Combinatorics;
- 68R10
- E-Print:
- 20 pages