Colouring Non-Even Digraphs

Millani, Marcelo Garlet; Steiner, Raphaël; Wiederrecht, Sebastian

doi:10.48550/arxiv.1903.02872

Cited by 3 publications

(7 citation statements)

References 19 publications

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“…It has become apparent that the dichromatic number acts as a natural directed counterpart of the chromatic number of an undirected graph. Numerous recent results (see [AH15], [MW16], [ACH + 16], [LM17], [BHKL18], [HLTW19], [MSW19]) support this claim. Formally, we consider the following problem.…”

Section: Dichromatic Numbermentioning

confidence: 86%

“…While graph colouring famously can be solved by an FPT-algorithm with respect to treewidth, it was shown in [MSW19] that even for bounded size of a directed feedback vertex set, deciding whether a directed graph has dichromatic number at most 2 is NP-complete. This rules out efficient parameterisations by most known directed width parameters such as directed treewidth, DAG-width or Kelly-width, as all of these are upper bounded in terms of the size of a smallest feedback vertex set.…”

Section: Digraph Colouring Short: DCmentioning

confidence: 99%

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Parameterized Algorithms for Directed Modular Width

Steiner

Wiederrecht

2020

Lecture Notes in Computer Science

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Many well-known NP-hard algorithmic problems on directed graphs resist efficient parametrisations with most known width measures for directed graphs, such as directed treewidth, DAG-width, Kelly-width and many others. While these focus on measuring how close a digraph is to an oriented tree resp. a directed acyclic graph, in this paper, we investigate directed modular width as a parameter, which is closer to the concept of clique-width. We investigate applications of modular decompositions of directed graphs to a wide range of algorithmic problems and derive FPT-algorithms for several well-known digraph-specific NP-hard problems, namely minimum (weight) directed feedback vertex set, minimum (weight) directed dominating set, digraph colouring, directed Hamiltonian path/cycle, partitioning into paths, (capacitated) vertex-disjoint directed paths, and the directed subgraph homeomorphism problem. The latter yields a polynomial-time algorithm for detecting topological minors in digraphs of bounded directed modular width. Finally we illustrate that also other structural digraph parameters, such as directed pathwidth and the cycle-rank can be computed efficiently using directed modular width as a parameter.

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Section: Dichromatic Numbermentioning

confidence: 86%

Section: Digraph Colouring Short: DCmentioning

confidence: 99%

Parameterized Algorithms for Directed Modular Width

Steiner

Wiederrecht

2020

Lecture Notes in Computer Science

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“…Here we consider three popular variants: strong minors, butterfly minors, and topological minors. The containment of these different minors in dense digraphs as well as their relation to the dichromatic number has already been studied in several previous works, see, for example, [2,12,15] for strong minors, [5,11,16,20] for butterfly minors, and [1,6,7,[17][18][19]25] for topological minors. Given digraphs D and H , we say that D is a strong H -minor model if V D ( ) can be partitioned into nonempty sets ∈ X v V H { :…”

Section: Introductionmentioning

confidence: 99%

Complete directed minors and chromatic number

Mészáros

Steiner

2022

Journal of Graph Theory

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The dichromatic number → χ D ( ) of a digraph D is the smallest k for which it admits a k-coloring where every color class induces an acyclic subgraph. Inspired by Hadwiger's conjecture for undirected graphs, several groups of authors have recently studied the containment of complete directed minors in digraphs with a given dichromatic number. In this note we exhibit a relation of these problems to Hadwiger's conjecture. Exploiting this relation, we show that every directed graph excluding the complete digraph ↔ K t of order t as a strong minor or as a butterfly minor is O t t ( (log log ) ) 6colorable. This answers a question by Axenovich, Girão, Snyder, and Weber, who proved an upper bound of t4 t for the same problem. A further consequence of our results is that every digraph of dichromatic number n 22 contains a subdivision of every n-vertex subcubic digraph, which makes progress on a set of problems raised by Aboulker, Cohen, Havet, Lochet, Moura, and Thomassé.

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“…The containment of these different minors in dense digraphs as well as their relation to the dichromatic number have already been studied in several previous works, see e.g. [2,8,11] for strong minors, [3,7,12,16] for butterfly minors and [1,4,5,13,14,15,20] for topological minors.…”

Section: Introductionmentioning

confidence: 99%

“…In [16] as a butterfly minor, since its dichromatic number exceeds k. Therefore, for every k we have…”

Section: Introductionmentioning

confidence: 99%

Complete minors in digraphs with given dichromatic number

Mészáros,

Steiner

2021

Preprint

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The dichromatic number χ(D) of a digraph D is the smallest k for which it admits a k-coloring where every color class induces an acyclic subgraph. Inspired by Hadwiger's conjecture for undirected graphs, several groups of authors have recently studied the containment of directed graph minors in digraphs with given dichromatic number. In this short note we improve several of the existing bounds and prove almost linear bounds by reducing the problem to a recent result of Postle on Hadwiger's conjecture.

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Colouring Non-Even Digraphs

Cited by 3 publications

References 19 publications

Parameterized Algorithms for Directed Modular Width

Parameterized Algorithms for Directed Modular Width

Complete directed minors and chromatic number

Complete minors in digraphs with given dichromatic number

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