Directed NLC-width

Gurski, Frank; Wanke, Egon; Yilmaz, Eda

doi:10.1016/j.tcs.2015.11.003

Cited by 24 publications

(23 citation statements)

References 25 publications

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“…It also remains to generalize the shown results for oriented coloring on oriented graphs of bounded directed clique-width as given in [9,22]. By the existence of a monadic second order logic formula it follows that for every c the problem OCN c is fixed parameter tractable w.r.t.…”

Section: Discussionmentioning

confidence: 85%

“…the directed clique-width of the input graph is still open. Since the directed clique-width of a digraph is always greater or equal the undirected clique-width of the corresponding underlying undirected graph [22], the result of Ganian [15] (see also Section 1) does not imply a hardness result.…”

Section: Discussionmentioning

confidence: 99%

“…Further, we give a linear time algorithm for the graph isomorphism problem on oriented co-graphs. Since oriented co-graphs have a directed NLC-width of one [22], our results provide a useful basis for exploring the complexity of OCN related to width parameters (cf. Section 7).…”

Section: Introductionmentioning

confidence: 99%

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Oriented Coloring on Recursively Defined Digraphs

2019

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Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented graph G = (V, A) is a partition of the vertex set V into k independent sets such that all the arcs linking two of these subsets have the same direction. The oriented chromatic number of an oriented graph G is the smallest k such that G allows an oriented k-coloring. Deciding whether an acyclic digraph allows an oriented 4-coloring is NP-hard. It follows that finding the chromatic number of an oriented graph is an NP-hard problem, too. This motivates to consider the problem on oriented co-graphs. After giving several characterizations for this graph class, we show a linear time algorithm which computes an optimal oriented coloring for an oriented co-graph. We further prove how the oriented chromatic number can be computed for the disjoint union and order composition from the oriented chromatic number of the involved oriented cographs. It turns out that within oriented co-graphs the oriented chromatic number is equal to the length of a longest oriented path plus one. We also show that the graph isomorphism problem on oriented co-graphs can be solved in linear time.

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Section: Discussionmentioning

confidence: 85%

Section: Discussionmentioning

confidence: 99%

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Oriented Coloring on Recursively Defined Digraphs

2019

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“…In [18] it has been shown that directed co-graphs can be characterized by the eight forbidden induced subdigraphs shown in Table 3. In [38] the relation of directed co-graphs to the set of graphs of directed NLC-width 1 and to the set of graphs of directed clique-width 2 is analyzed.…”

Section: Equivalent Parametersmentioning

confidence: 99%

Comparing Linear Width Parameters for Directed Graphs

Gurski

Rehs

2019

Theory Comput Syst

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In this paper we introduce the linear clique-width, linear NLC-width, neighbourhood-width, and linear rank-width for directed graphs. We compare these parameters with each other as well as with the previously defined parameters directed path-width and directed cut-width. It turns out that the parameters directed linear clique-width, directed linear NLC-width, directed neighbourhood-width, and directed linear rank-width are equivalent in that sense, that all of these parameters can be upper bounded by each of the others. For the restriction to digraphs of bounded vertex degree directed path-width and directed cut-width are equivalent. Further for the restriction to semicomplete digraphs of bounded vertex degree all six mentioned width parameters are equivalent. We also show close relations of the measures to their undirected versions of the underlying undirected graphs, which allow us to show the hardness of computing the considered linear width parameters for directed graphs. Further we give first characterizations for directed graphs defined by parameters of small width.

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“…Among these are directed co-graphs, which can be defined by disjoint union, order composition, and series composition [13]. Directed co-graphs are useful to characterize digraphs of directed NLC-width 1 and digraphs of directed clique-width 2 [19] and are useful for the reconstruction of the evolutionary history of genes or species using genomic sequence data [23,33]. Our results imply the equality of directed path-width and directed tree-width for directed co-graphs and a linear-time solution for computing the directed path-width and directed tree-width of directed co-graphs.…”

Section: Introductionmentioning

confidence: 99%

Forbidden Directed Minors, Directed Path-Width and Directed Tree-Width of Tree-Like Digraphs

Gurski

Rehs

2019

Lecture Notes in Computer Science

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In this paper we consider the directed path-width and directed tree-width of recursively defined digraphs. As an important combinatorial tool, we show how the directed path-width and the directed tree-width can be computed for the disjoint union, order composition, directed union, and series composition of two directed graphs. These results imply the equality of directed path-width and directed tree-width for all digraphs which can be defined by these four operations. This allows us to show a linear-time solution for computing the directed path-width and directed tree-width of all these digraphs. Since directed co-graphs are precisely those digraphs which can be defined by the disjoint union, order composition, and series composition our results imply the equality of directed path-width and directed tree-width for directed co-graphs and also a linear-time solution for computing the directed path-width and directed tree-width of directed co-graphs, which generalizes the known results for undirected co-graphs of Bodlaender and Möhring.We assume that V G ⊆ X e for every e ∈ E T . We define (Whenever this leads to an empty set W ′ s where t is the predecessor of s in T ′ J we remove vertex s from T ′ J and replace every arc (s, t ′ ) by (t, t ′ ) with the corresponding set X (t,t ′ ) = X (s,t ′ ) ∩ V H .is a directed tree-decomposition of H as follows.• W ′ J is a partition of V H into nonempty sets. • Let e be an arc in T ′ J which is also in T J . Since e ∼ s implies W s = W ′ s = {v} for some v ∈ V H normality condition remains true. Arcs (t, t ′ ) in T ′ J which are not in T J are obtained by two arcs (t, s) and (s, t ′ ) from T J . If ∪{W r | r ∈ V T , t ′ ≤ r} is X (s,t ′ ) -normal, then ∪{W r | r ∈ V ′ T , t ′ ≤ r} is X (t,t ′ ) -normal since X (t,t ′ ) = X (s,t ′ ) ∩ V H .

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Directed NLC-width

Cited by 24 publications

References 25 publications

Oriented Coloring on Recursively Defined Digraphs

Oriented Coloring on Recursively Defined Digraphs

Comparing Linear Width Parameters for Directed Graphs

Forbidden Directed Minors, Directed Path-Width and Directed Tree-Width of Tree-Like Digraphs

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