On the Parameterized Complexity of Colorful Components and Related Problems

Misra, Neeldhara

doi:10.1007/978-3-319-94667-2_20

Cited by 4 publications

(3 citation statements)

References 16 publications

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“…We now prove two FPT results for two different parameters. Our proof for the next result uses similar arguments to the proof sketch of Theorem 8 given in [13]. However, the details of both proofs are different, as optimal solutions for Colourful Partition do not necessarily translate into optimal solutions for Colourful Components.…”

Section: Parameterized Complexitymentioning

confidence: 96%

Finding a Small Number of Colourful Components

Bulteau,

Dabrowski,

Fertin

et al. 2018

Preprint

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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.

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Section: Parameterized Complexitymentioning

confidence: 96%

Finding a Small Number of Colourful Components

Bulteau,

Dabrowski,

Fertin

et al. 2018

Preprint

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“…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).…”

Section: Introductionmentioning

confidence: 99%

“…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]. Now, we give the formal definitions of the problems considered herein.…”

Section: Introductionmentioning

confidence: 99%

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

Chlebíková

Dallard

2019

Lecture Notes in Computer Science

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A connected component of a vertex-coloured graph is said to be colourful if all its vertices have different colours, and a graph is colourful if all its connected components are colourful. Given a vertex-coloured graph, the Colourful Components problem asks whether there exist at most p edges whose removal makes the graph colourful, and the Colourful Partition problem asks whether there exists a partition of the vertex set with at most p parts such that each part induces a colourful component. We study the problems on k-caterpillars (caterpillars with hairs of length at most k) and explore the boundary between polynomial and NP-complete cases. It is known that the problems are NP-complete on 2-caterpillars with unbounded maximum degree. We prove that both problems remain NP-complete on binary 4-caterpillars and on ternary 3caterpillars. This answers an open question regarding the complexity of the problems on trees with maximum degree at most 5. On the positive side, we give a linear time algorithm for 1-caterpillars with unbounded degree, even if the backbone is a cycle, which outperforms the previous best complexity for paths and widens the class of graphs. Finally, we answer an open question regarding the complexity of Colourful Components on graphs with maximum degree at most 5. We show that the problem is NP-complete on 5-coloured planar graphs with maximum degree 4, and on 12-coloured planar graphs with maximum degree 3. Since the problem can be solved in polynomial-time on graphs with maximum degree 2, the results are the best possible with regard to the maximum degree.

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