Finding a Small Number of Colourful Components

TL;DR

This paper investigates the computational complexity of colourful partition problems in graphs, providing new hardness results and a comprehensive parameterized analysis to distinguish between tractable and NP-hard cases.

Contribution

It offers new hardness results and a detailed parameterized complexity study for the colourful partition and components problems, clarifying their computational boundaries.

Findings

NP-hardness results for various instances

Identification of tractable cases under certain parameters

Complete parameterized complexity classification

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.