Finding a Small Number of Colourful Components

Abstract

A partition (V 1 , . . . , 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.

Towards a Complexity Dichotomy for Colourful Components Problems on k-caterpillars and Small-Degree Planar Graphs

“…In Section 2.1, we prove that Colourful Components and Colourful Partition are NP-complete on binary 4-caterpillars and on ternary 3-caterpillar, hence with the maximum degree at most 3 or 4. This answers an open question, proposed in [5], regarding the complexity of the problems on trees with maximum degree at most 5. Nonetheless, we propose a linear time algorithm for Colourful Components and Colourful Partition on 1-caterpillars and cyclic 1-caterpillars with unbounded degree in Section 2.2.…”

“…Observe that, on a tree, there is a solution to Colourful Components with p edges if and only if there is a solution to Colourful Partition with p + 1 parts. However, this is not the case on general graphs [5]. Both problems are known to be NP-complete on subdivided stars [6], trees of diameter at most 4 [4], and trees with maximum degree 6 [5].…”

“…However, this is not the case on general graphs [5]. Both problems are known to be NP-complete on subdivided stars [6], trees of diameter at most 4 [4], and trees with maximum degree 6 [5]. Trees of diameter at most 4 are in fact a subclass of 2-caterpillars, so both problems are NP-complete on 2-caterpillars when the maximum degree is unbounded.…”

“…The quality of a partition of a set of genes into orthologous genes can be expressed in different ways. Minimising the number of similar genes in different subsets of the partition is a well studied variant [4,5,8,13,15], and it corresponds to minimising the number of edges between the colourful components (as in Colourful Components). Alternatively, one can ask for a partition of minimum size, i.e which contains the minimum number of orthologous genes, or equivalently the minimum number of colourful components [1,5,6] (as in Colourful Partition).…”

“…Minimising the number of similar genes in different subsets of the partition is a well studied variant [4,5,8,13,15], and it corresponds to minimising the number of edges between the colourful components (as in Colourful Components). Alternatively, one can ask for a partition of minimum size, i.e which contains the minimum number of orthologous genes, or equivalently the minimum number of colourful components [1,5,6] (as in Colourful Partition). Another variant, not studied in this paper, considers the objective function that maximises the number of edges in the transitive closure [1,6,13].…”