Finding a Small Number of Colourful Components

Abstract : A partition $(V_1,\ldots,V_k)$ of the vertex set of a graph $G$ with a (not necessarily proper) colouring $c$ is colourful if no two vertices in any $V_i$ have the same colour and every set $V_i$ induces a connected graph. The COLOURFUL PARTITION problem is to decide whether a coloured graph $(G,c)$ has a colourful partition of size at most $k$. 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 $p$ 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.
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Contributor : Stéphane Vialette <>
Submitted on : Saturday, January 19, 2019 - 9:46:34 AM
Last modification on : Friday, June 21, 2019 - 10:28:00 AM

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  • HAL Id : hal-01986725, version 1
  • ARXIV : 1808.03561


Laurent Bulteau, Konrad K. Dabrowski, Guillaume Fertin, Matthew Johnson, Daniel Paulusma, et al.. Finding a Small Number of Colourful Components. 2019. ⟨hal-01986725⟩



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