Finding a Small Number of Colourful Components
Abstract
A partition of the vertex set of a graph with a (not necessarily proper) colouring is colourful if no two vertices in any have the same colour and every set induces a connected graph. The COLOURFUL PARTITION problem is to decide whether a coloured graph has a colourful partition of size at most . This problem is closely related to the COLOURFUL COMPONENTS problem, which is to decide whether a graph can be modified into a graph whose connected components form a colourful partition by deleting at most edges. Nevertheless we show that COLOURFUL PARTITION and COLOURFUL COMPONENTS may have different complexities for restricted instances. We tighten known NP-hardness results for both problems and in addition we prove new hardness and tractability results for COLOURFUL PARTITION. Using these results we complete our paper with a thorough parameterized study of COLOURFUL PARTITION.
Cite
@article{arxiv.1808.03561,
title = {Finding a Small Number of Colourful Components},
author = {Laurent Bulteau and Konrad K. Dabrowski and Guillaume Fertin and Matthew Johnson and Daniel Paulusma and Stephane Vialette},
journal= {arXiv preprint arXiv:1808.03561},
year = {2018}
}