Parameterized Algorithms for Directed Modular Width

Steiner, Raphael; Wiederrecht, Sebastian

doi:10.1007/978-3-030-39219-2_33

Cited by 8 publications

(7 citation statements)

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“…the set of all oriented paths { − → P n | n ≥ 1}, the set of all oriented cycles { − → C n | n ≥ 1}, and the set of all msp-digraphs. An advantage of directed modular width is that it can be computed efficiently [SW19,SW20]. Lemma 6.16 and Corollary 6.12 lead to the following result.…”

Section: Disjoint Union Of Two Labeled Digraphs G and H (Denoted By G...mentioning

confidence: 93%

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Efficient computation of the oriented chromatic number of recursively defined digraphs

Gurski¹,

Komander²,

Lindemann³

2020

Preprint

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In this paper we consider colorings of oriented graphs, i.e. digraphs without cycles of length 2. Given some oriented graph G = (V, E), an oriented r-coloring for G is a partition of the vertex set V into r independent sets, such that all the arcs between two of these sets have the same direction. The oriented chromatic number of G is the smallest integer r such that G permits an oriented r-coloring. Deciding whether an acyclic digraph has an oriented 4-coloring is NP-hard, which motivates to consider the problem on special graph classes.In this paper we consider the Oriented Chromatic Number problem on classes of recursively defined oriented graphs. Oriented co-graphs (short for oriented complement reducible graphs) can be recursively defined defined from the single vertex graph by applying the disjoint union and order composition. This recursive structure allows to compute an optimal oriented coloring and the oriented chromatic number in linear time. We generalize this result using the concept of perfect orderable graphs. Therefore, we show that for acyclic transitive digraphs every greedy coloring along a topological ordering leads to an optimal oriented coloring. Msp-digraphs (short for minimal series-parallel digraphs) can be defined from the single vertex graph by applying the parallel composition and series composition. We prove an upper bound of 7 for the oriented chromatic number for msp-digraphs and we give an example to show that this is bound best possible. We apply this bound and the recursive structure of msp-digraphs to obtain a linear time solution for computing the oriented chromatic number of msp-digraphs.In order to generalize the results on computing the oriented chromatic number of special graph classes, we consider the parameterized complexity of the Oriented Chromatic Number problem by so-called structural parameters, which are measuring the difficulty of decomposing a graph into a special tree-structure.

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Section: Disjoint Union Of Two Labeled Digraphs G and H (Denoted By G...mentioning

confidence: 93%

“…the parameter directed modular-width (dmw). Directed modular-width was introduced and applied to the Acyclic Chromatic Number problem in [SW19,SW20].…”

Section: Parameterization By Directed Modular-widthmentioning

confidence: 99%

Efficient computation of the oriented chromatic number of recursively defined digraphs

Gurski¹,

Komander²,

Lindemann³

2020

Preprint

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“…An acyclic r-coloring of a digraph G = (V, E) is a partition of the vertex set V into r sets such that all sets induce an acyclic subdigraph in G. The dichromatic number of G is the smallest r such that G has an acyclic r-coloring. Acyclic colorings of digraphs received a lot of attention in [BFJ + 04, Moh03,NL82] and also in recent works [LM17,MSW19,SW20]. The dichromatic number is one of two basic concepts for the class of perfect digraphs [AH15] and can be regarded as a natural counterpart of the well known chromatic number for undirected graphs.…”

Section: Introductionmentioning

confidence: 99%

“…This result implies that there are no XP-algorithms 1 for the Dichromatic Number problem parameterized by directed width parameters such as directed path-width, directed treewidth, DAG-width or Kelly-width, since all of these are upper bounded in terms of the size of a smallest feedback vertex set. The first positive result concerning structural parameterizations of the Dichromatic Number problem is the existence of an FPT-algorithm 2 for the Dichromatic Number problem parameterized by directed modular width [SW19].…”

Section: Introductionmentioning

confidence: 99%

“…Since the directed clique-width of a digraph is at most its directed modular width [SW20], we reprove the existence of an XP-algorithm for DCN and an FPT-algorithm for DCN r parameterized by directed modular width [SW19]. On the other hand, there exist several classes of digraphs of bounded directed clique-width and unbounded directed modular width, which implies that directed clique-width is the more powerful parameter and that the results of [SW19] does not imply any parameterized algorithm for directed clique-width.…”

Section: Introductionmentioning

confidence: 99%

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Acyclic coloring of special digraphs

Gurski,

Komander,

Rehs

2020

Preprint

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An acyclic r-coloring of a directed graph G = (V, E) is a partition of the vertex set V into r acyclic sets. The dichromatic number of a directed graph G is the smallest r such that G allows an acyclic r-coloring. For symmetric digraphs the dichromatic number equals the well-known chromatic number of the underlying undirected graph. This allows us to carry over the W[1]-hardness and lower bounds for running times of the chromatic number problem parameterized by clique-width to the dichromatic number problem parameterized by directed clique-width. We introduce the first polynomial-time algorithm for the acyclic coloring problem on digraphs of constant directed clique-width. From a parameterized point of view our algorithm shows that the Dichromatic Number problem is in XP when parameterized by directed clique-width and extends the only known structural parameterization by directed modular width for this problem. For directed co-graphs, which is a class of digraphs of directed clique-width 2, and several generalizations we even show linear time solutions for computing the dichromatic number. Furthermore, we conclude that directed co-graphs and the considered generalizations lead to subclasses of perfect digraphs. For directed cactus forests, which is a set of digraphs of directed tree-width 1, we conclude an upper bound of 2 for the dichromatic number and we show that an optimal acyclic coloring can be computed in linear time.

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