Oriented coloring on recursively defined digraphs

04/02/2019 ∙ by Frank Gurski, et al. ∙ 0

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. 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 co-graphs. 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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1 Introduction

Graph coloring is one of the basic problems in graph theory, which has already been considered in the 19th century. A -coloring for an undirected graph is a -labelling of the vertices of such that no two adjacent vertices have the same label. The smallest such that a graph has a -coloring is named the chromatic number of . As even the problem, if a graph has a -coloring or not, is NP-complete, finding the chromatic number of an undirected graph is an NP-hard problem. But there are many efficient solutions for the coloring problem on special graph classes, like chordal graphs [15], comparability graphs [23], and co-graphs [6].

Oriented coloring has been introduced much later by Courcelle [7]. One could easily apply the definition of graph coloring to directed graphs, but as this would not take the direction of the arcs into account, this is not very interesting. For such a definition, the coloring of a directed graph would be the coloring of the underlying undirected graph.

Oriented coloring considers also the direction of the arcs. An oriented -coloring of an oriented graph is a partition of the vertex set into independent sets such that all the arcs linking two of these subsets have the same direction. In the oriented chromatic number problem (OCN for short) it is given some oriented graph and some integer and it is to decide whether there is an oriented -coloring for . Even the restricted problem if is constant and does not belong to the input ( for short) is hard. is NP-complete even for DAGs [10], whereas the undirected problem is easy for trees.

Right now, the definition of oriented colorings is mostly considered for undirected graphs. There the maximum value of all possible orientations of an undirected graph is considered. For several special undirected graph classes the oriented chromatic number has been bounded. Among these are outerplanar graphs [28], planar graphs [26], and Halin graphs [11]. Ganian has shown in [14] an FPT-algorithm for OCN w.r.t. the parameter tree-width (of the underlying undirected graph). Further he has shown that OCN is DET-hard for classes of oriented graphs such that the underlying undirected class has bounded rank-width.

Figure 1: Special oriented graphs: oriented cycle and transitive tournament .

Oriented colorings of special digraph classes seem to be uninvestigated up to now. The main reason is that the oriented chromatic number of the disjoint union of two oriented graphs can be larger than the maximum oriented chromatic number of the involved graphs (cf. Figure 1 and Example 4.7). In this paper we consider oriented coloring problem restricted to oriented co-graphs, which are obtained from directed co-graphs [4] by omitting the series operation. Oriented co-graphs were already analyzed by Lawler in [25] and [6, Section 5] using the notation of transitive series parallel (TSP) digraphs. We give several characterizations for oriented co-graphs and show that for oriented co-graphs, the oriented chromatic number of the disjoint union of oriented graphs is equal to the maximum oriented chromatic number of the involved graphs. Further, we show that for every oriented graph the oriented chromatic number of the order composition of oriented graphs is equal to the sum of the oriented chromatic number of the involved graphs. To show this we introduce an algorithm that computes an optimal oriented coloring and thus the oriented chromatic number of oriented co-graphs in linear time. We also consider the longest oriented path problem on oriented co-graphs. It turns out that within oriented co-graphs the oriented chromatic number is equal to the length of a longest oriented path plus one. Further, we give a linear time algorithm for the graph isomorphism problem on oriented co-graphs. Our results provide a useful basis for exploring the complexity of OCN related to width parameters.

2 Preliminaries

2.1 Graphs and Digraphs

We use the notations of Bang-Jensen and Gutin [2] for graphs and digraphs.

For some given digraph , we define its underlying undirected graph by ignoring the directions of the edges, i.e. and for some class of digraphs , let . For some (di)graph class we define as the set of all (di)graphs such that no induced sub(di)graph of is isomorphic to a member of .

An oriented graph is a digraph with no loops and no opposite arcs. We recall some special oriented graphs. By

, we denote the oriented path on vertices, by

, we denote the oriented cycle on vertices and by we denote the transitive tournament on vertices.

2.2 Undirected Co-Graphs

Let be vertex-disjoint graphs.

  • The disjoint union of , denoted by , is the graph with vertex set and edge set .

  • The join composition of , denoted by , is defined by their disjoint union plus all possible edges between vertices of and for all , .

The set of all graphs which can be defined by the disjoint union and join composition is characterized as the set of all co-graphs. It is well known that co-graphs are precisely the -free graphs [6].

2.3 Undirected Graph Coloring

Definition 2.1 (Graph Coloring)

A -coloring of a graph is a mapping such that:

  • for every

The chromatic number of , denoted by , is the smallest such that has a -coloring.

On undirected co-graphs, the graph coloring problem is easy to solve by the following result proved by Corneil et al.:

Lemma 2.2 ([6])

Let be vertex-disjoint graphs.

Proposition 2.3

Let be a co-graph, then can be computed in linear time.

The undirected coloring problem, i.e. computing , can be solved by an FPT-algorithm w.r.t. the tree-width of the input graph [17]. In contrast, this is not true for clique-width, since in [13] it has been shown that the undirected coloring problem is -hard w.r.t. the clique-width of the input graph. That is, under reasonable assumptions an XP-algorithm is the best one can hope for and such algorithms are known [12].

2.4 Directed Co-Graphs

The following operations for digraphs have already been considered by Bechet et al. in [4]. Let be vertex-disjoint digraphs.

  • The disjoint union of , denoted by , is the digraph with vertex set and arc set .

  • The series composition of , denoted by , is defined by their disjoint union plus all possible arcs between vertices of and for all , .

  • The order composition of , denoted by , is defined by their disjoint union plus all possible arcs from vertices of to vertices of for all .

The set of all digraphs which can be defined by the disjoint union, series composition, and order composition is characterized as the set of all directed co-graphs [4]. Obviously for every directed co-graph we can define a tree structure, denoted as the di-co-tree. The leaves of the di-co-tree represent the vertices of the graph and the inner nodes of the di-co-tree correspond to the operations applied on the subexpressions defined by the subtrees. For every directed co-graph one can construct a di-co-tree in linear time, see [9].

In [3] it is shown that the weak -linkage problem can be solved in polynomial time for directed co-graphs. By the recursive structure there exist dynamic programming algorithms to compute the size of a largest edgeless subdigraph, the size of a largest subdigraph which is a tournament, the size of a largest semicomplete subdigraph, and the size of a largest complete subdigraph for every directed co-graph in linear time. Also the Hamiltonian path, Hamiltonian cycle, regular subdigraph, and directed cut problem are polynomial on directed co-graphs [18]. Calculs of directed co-graphs were also considered in connection with pomset logic in [27]. Further the directed path-width and also the directed tree-width can be computed in linear time for directed co-graphs [20, 19].

In [9] it has been shown that directed co-graphs can be characterized by the eight forbidden induced subdigraphs shown in Figure 2.

Figure 2: The eight forbidden induced subdigraphs for directed co-graphs.

3 Oriented Co-Graphs

Oriented colorings are defined on oriented graphs, which are digraphs with no bidirected edges. Therefor we introduce oriented co-graphs by restricting directed co-graphs from [4] by omitting the series operation.

Definition 3.1 (Oriented Co-Graphs)

The class of oriented co-graphs is recursively defined as follows.

  1. Every digraph on a single vertex , denoted by , is an oriented co-graph.

  2. If are vertex-disjoint oriented co-graphs, then

    1. and

    2. are oriented co-graphs.

The class of oriented co-graphs was already analyzed by Lawler in [25] and [6, Section 5] using the notation of transitive series parallel (TSP) digraphs. A digraph is called transitive if for every pair and of arcs with the arc also belongs to .

Theorem 3.2 ([6])

A graph is a co-graph if and only if there exists an orientation of such that is an oriented co-graph.

A di-co-tree is canonical if on every path from the root to the leaves of , the labels disjoint union and order operation strictly alternate. Since the disjoint union and the order composition are associative we always can restrict to canonical di-co-trees.

Lemma 3.3

Let be an oriented co-graph and be a di-co-tree for . Then can be transformed in linear time into a canonical di-co-tree for .

The recursive definitions of oriented and undirected co-graphs lead to the following observation.

Observation 3.4

For every oriented co-graph the underlying undirected graph is a co-graph.

The reverse direction of this observation only holds under certain conditions, see Theorem 3.6. By we denote the complete biorientation of a path on vertices.

Lemma 3.5

Let be a digraph such that , then is transitive.

Proof  Let be two arcs of . Since we know that . Further since , we know that and are connected either only by or by and , which implies that is transitive.

Oriented co-graphs can be characterized by forbidden subdigraphs as follows.

Theorem 3.6

Let be a digraph. The following properties are equivalent:

  1. is an oriented co-graph.

  2. .

  3. and .

  4. and is a co-graph.

  5. has directed NLC-width and .

  6. has directed clique-width at most and .

  7. is transitive and .

Proof  If is an oriented co-graph, then is a directed co-graph and by [9] it holds . Furthermore because of the missing series composition. This leads to . If then and is a directed co-graph. Since there is no series operation in any construction of which implies that is an oriented co-graph. Since co-graphs are precisely the -free graphs [6]. By Lemma 3.5. If is transitive, then . and By [21]. By Observation 3.4. If does not contain a , then can not contain any orientation of .

Among others are two subclasses of oriented co-graphs, which will be of interest within our results. By restricting within Definition 3.1 (2) to and graph or to an edgeless graph or to a single vertex, we obtain the class of all oriented simple co-graphs or oriented threshold graphs, respectively. The class of oriented threshold graphs has been introduced by Boeckner in [5].

4 Graph Coloring on Recursively Defined Digraphs

4.1 Oriented Graph Coloring Problem

Oriented graph coloring has been introduced by Courcelle [7] in 1994. Most results on this problem are for orientations of undirected graphs. Now we consider oriented graph coloring on recursively defined oriented graph classes.

Definition 4.1 (Oriented Graph Coloring [7])

An oriented -coloring of an oriented graph is a mapping such that:

  • for every

  • for every two arcs and with

The oriented chromatic number of , denoted by , is the smallest such that has an oriented -coloring. The vertex sets , , divide a partition of into so called color classes.

For two oriented graphs and a homomorphism from to , for short, is a mapping , such that implies that . The oriented graphs and are homomorphically equivalent, if there is a homomorphism from to and there is one homomorphism from to . A homomorphism from to can be regarded as an oriented coloring of that uses the vertices of as colors classes. This leads to equivalent definitions of oriented colorings and oriented chromatic numbers. There is an oriented -coloring of an oriented graph if and only if there is a homomorphism from to some oriented graph on vertices. That is, the oriented chromatic number of is the minimum number of vertices in an oriented graph such that there is a homomorphism from to . Obviously, can be chosen as a tournament.

Observation 4.2

There is an oriented -coloring of an oriented graph if and only if there is a homomorphism from to some tournament on vertices. Further the oriented chromatic number of is the minimum number of vertices in a tournament such that there is a homomorphism from to .

Lemma 4.3

Let be an oriented graph and be a subdigraph of , then .

Example 4.4

For oriented paths and oriented cycles we know: , , ,

An oriented graph is an oriented clique (o-clique) if . Thus all graphs given in Example 4.4 are oriented cliques.

  • Oriented Chromatic Number (OCN)

  • An oriented graph and a positive integer .

  • Is there an oriented -coloring for ?

If is constant and is not part of the input, the corresponding problem is denoted by . is NP-complete even for DAGs [10].

The definition of oriented colorings is also used for undirected graphs . There the maximum value of all possible orientations of is considered. In this sense, every tree has oriented chromatic number at most . For several further graph classes there are bounds on the oriented number. Among these are outerplanar graphs [28], planar graphs [26], and Halin graphs [11].

4.2 Oriented Graph Coloring for Oriented Graphs

Oriented graph coloring has not yet been considered for recursively defined graphs, though it has been analyzed for some graph operations. In this section we show some results of oriented graph coloring on some graph operations and conclude algorithms for some recursively defined oriented graph classes. This will also be useful for the following section.

First we give some results on the oriented graph coloring for general recursively defined oriented graphs. These results will be very useful to prove our results for oriented co-graphs in the next section.

Lemma 4.5

Let be vertex-disjoint oriented graphs.

Proof 

  1. Since no new arcs are inserted can keep its colors. The added isolated vertex gets a color of in order to get a valid coloring for .

    Since is an induced subdigraph of this relation holds by Lemma 4.3.

  2. Since the digraphs are induced subdigraphs of digraph , all values lead to a lower bound on the number of necessary colors for the combined graph by Lemma 4.3.

  3. For let and be a coloring for . For we define a mapping as follows.

    This mapping satisfies the definition of an oriented coloring as follows. No two adjacent vertices from , have the same color by assumption and by our definition of mapping . A vertex of and a vertex of for are always adjacent but are never colored equally by our definition of mapping .

    Further the arcs between two color classes of every , have the same direction by our definition of mapping . The arcs between a color class of and a color class of for have the same direction by the definition of the order operation.

    Since every , , is an induced subdigraph of the combined graph, all values lead to a lower bound on the number of necessary colors for the combined graph by Lemma 4.3. Further the order operations implies that for every no vertex in can be colored in the same way as a vertex in . Thus leads to a lower bound on the number of necessary colors for the combined graph.

This shows the statements of the lemma.

By Lemma 4.5 we can solve oriented coloring for oriented simple co-graphs and thus also for subclasses such as oriented threshold graphs and transitive tournaments in linear time.

Proposition 4.6

Let be an oriented simple co-graph, then and all values can be computed in linear time.

It is not easy to generalize these results to oriented co-graphs. To do so, we would need to compute the oriented chromatic number of the disjoint union of two oriented co-graphs with at least two vertices. But it is not possible to compute this oriented chromatic number of the disjoint union of general oriented graphs from the oriented chromatic numbers of the involved graphs. In Lemma 4.5 (2) we only show a lower bound. The following example proves that in general this can not be strengthened to equality.

Example 4.7

The two graphs and in Figure 1 have the same oriented chromatic number , but their disjoint union needs more colors.

On the other hand, there are several examples for which the disjoint union does not need more than colors, such as for the union of two isomorphic oriented graphs. By Theorem 3.6, we know that shown in Figure 1 is an oriented co-graph but shown in Figure 1 is not an oriented co-graph. So the question arises whether oriented coloring could be closed under disjoint union if restricted to oriented co-graphs.

4.3 Oriented Graph Coloring for Oriented Co-Graphs

In order to solve OCN restricted to oriented co-graphs we created an algorithm, which is shown in Figure 3. The method traverses a canonical di-co-tree for using a depth-first search such that for very inner vertex the children are visited from left to right. For every inner vertex of we store two values and . These values ensure that the vertices of corresponding to the leaves of the subtree rooted at will we labeled by labels such that . For every leaf vertex of we additionally store the label of the corresponding vertex of in . These values lead to an optimal oriented coloring of by the next theorem.

 Algorithm Label   if ( is a leaf of ) { ; ; ; } else { ; ; for all children of from left to right do { Label; if () ; if ( corresponds to a disjoint union) ; else corresponds to an order operation ; } } return ;  

Figure 3: Computing an oriented coloring of an oriented co-graph.
Theorem 4.8

Let be an oriented co-graph, then an optimal oriented coloring for and can be computed in linear time.

Proof  Let be an oriented co-graph. Using the method of [9] we can build a di-co-tree with root for in linear time. Further we can assume that is a canonical di-co-tree by Lemma 3.3. For some node of we define by the subtree of rooted at and by the subgraph of which is defined by . Obviously, for every vertex of the tree is a di-co-tree for the digraph which is also an oriented co-graph.

Next we verify that algorithm Label, shown in Figure 3, returns the value and even computes an oriented coloring for within array . Therefor we recursively show for every vertex of that after performing Label for all leaves of the value leads to an oriented coloring of using the colors 111Please note that using colors starting at values greater than 1 is not a contradiction to Definition 4.1. and the value leads to the oriented chromatic number of .

We distinguish the following three cases depending on the type of operation corresponding to the vertices of .

  • If is a leaf of then by the algorithm leads to an oriented coloring of .

    Further, which obviously corresponds to the oriented chromatic number of .

  • If is an inner vertex of which corresponds to an order operation and are the children of in .

    We already know that the oriented colorings of , , are feasible. Further by our algorithm a vertex from and a vertex from for are never colored equally in . The arcs between a color class of and a color class of for have the same direction by the definition of the order operation.

    By the algorithm, value is equal to . By Lemma 4.5 we conclude that is equal to .

  • If is an inner vertex of which corresponds to a disjoint union operation and are the children of in .

    We already know that the oriented colorings of , , are feasible. Since a disjoint union operation does not create any arcs, no two adjacent vertices have the same color in . Further, our method ensures that for every arc in it holds that . Thus all arcs between two color classes in have the same direction.

    By the algorithm value is equal to . By Lemma 4.5 we conclude that . The relation holds by the feasibility of our oriented coloring.

By applying the invariant for , the statements of the theorem follow.

Example 4.9

We illustrate the method given in Figure 3 by computing an oriented coloring for the oriented co-graph which is given by the canonical di-co-tree of Figure 4. On the left of each vertex of , the values and are given. An optimal oriented coloring for is given in blue letters below the leaves of . The root of leads to .

Figure 4: Canonical di-co-tree for oriented co-graph .

Next we can improve the result of Lemma 4.5(2) for oriented co-graphs.

Corollary 4.10

Let be vertex-disjoint oriented co-graphs. Then it holds

Proof  Let be an oriented co-graph and be a di-co-tree with root for . The method given in Figure 3 computes an oriented coloring using colors. Further the proof of Theorem 4.8 shows that .

Corollary 4.11

Let be an oriented co-graph. The following properties are equivalent:

  1. is an oriented clique.

  2. has a di-co-tree, which does not use any disjoint union operation.

  3. is a transitive tournament.

Further characterizations for transitive tournaments and thus for oriented co-graphs which are oriented cliques can be found in [16, Chapter 9].

As mentioned in Observation 4.2, oriented colorings of an oriented graph can be characterized by the existence of homomorphisms to tournaments. These tournaments are not necessarily transitive and is not necessarily homomorphically equivalent to some tournament. For oriented co-graphs we can show a deeper result.

Corollary 4.12

There is an oriented -coloring of an oriented co-graph if and only if there is a homomorphism from to some transitive tournament on vertices. Further the oriented chromatic number of an oriented co-graph is the minimum number such that is homomorphically equivalent with the transitive tournament .

Proof  Within an oriented co-graph the color classes of an oriented -coloring define a transitive tournament . If , then there is a homomorphism from to .

5 Longest Oriented Path for Oriented Graphs

  • Oriented Path (OP)

  • An oriented graph and a positive integer .

  • Is there an oriented path of length at least in ?

We can bound the path length using the oriented chromatic number when considering oriented co-graphs. Please note that in the subsequent results oriented path does not necessarily mean induced path (though, its definition implies no chord on the path).

Proposition 5.1 ([22])

A directed graph has a homomorphism to the transitive tournament if and only if there is no homomorphism of the oriented path to .

By Corollary 4.12 this leads to the next result.

Corollary 5.2

The oriented chromatic number of an oriented co-graph is the minimum number such that there is no homomorphism of the oriented path to .

In order to compute for some oriented graph the length of a longest oriented path , we give the next result.

Lemma 5.3

Let be vertex-disjoint oriented graphs.

Theorem 5.4

Let be an oriented co-graph, then the length of a longest oriented path can be computed in linear time.

Proposition 5.5

Let be an oriented co-graph, then .

Proof  The statement can be shown recursively on the structure of an oriented co-graph .

  • If , then it obviously holds .

  • If

  • If

This shows the statements of the lemma.

This implies that for every oriented co-graph the upper bound of Corollary 5.2 is tight.

Corollary 5.6

Let be an oriented co-graph, then and all values can be computed in linear time.

Proof  The first equality holds by Proposition 5.5 and remaining equalities follow by Lemma 2.2.

6 Graph Isomorphism for Oriented Co-Graphs

The isomorphism problem for undirected co-graphs has been shown to be solvable in polynomial time in [6]. This result can even be improved as follows. Two undirected co-graphs and are isomorphic if and only if their canonical co-trees and are isomorphic. A canonical co-tree for some co-graph can be found in linear time. Thus by applying a linear time isomorphism test for rooted labeled trees (cf. [1, Section 3.2]) on canonical co-trees for and one can decide in linear time, whether and are isomorphic.

We consider the corresponding problem for oriented co-graphs.

  • Oriented Co-Graph Isomorphism (OCI)

  • Two oriented co-graphs and .

  • Are and isomorphic, i.e. is there a bijection such that for all it holds if and only if ?

For oriented co-graphs, the method using an isomorphism test for rooted labeled trees on the co-trees does not work, since the order of the vertices, which are representing order operations in the di-co-tree, must be preserved. Our solution, given in Figure 5, is obtained by a modification of the method given in [1, Section 3.2]. W.l.o.g. we can assume that the height and the operation of the root of di-co-trees and are equal.

 Algorithm Test   let be the height of and for downto 0 do for all vertices on level in from left to right do if ( is a leaf) else let be the children of if ( corresponds to a union operation)

sort vector

ascending let be the sequence of all for all on level in for all vertices on level in from left to right do if ( is a leaf) else let be the children of if ( corresponds to a union operation) sort vector ascending let be the sequence of all for all on level in sort to obtain and sort to obtain if () return let be the set of all vectors on level in find a bijection for all vertices on level in do ; for all vertices on level in do ; return ;  

Figure 5: Testing graph isomorphism for two oriented co-graphs given by canonical di-co-trees.
Theorem 6.1

Let and be two oriented co-graphs, then oriented co-graph isomorphism for and can be solved in linear time.

7 Conclusions and Outlook

In this paper we have considered vertex coloring on oriented graphs. We were able to introduce linear time solutions for the oriented coloring problem, longest oriented path problem and isomorphism problem on oriented co-graphs. Our solutions are based on computations along a canonical di-co-tree for the given input co-graphs. Furthermore, it turns out that within oriented co-graphs, the oriented chromatic number is equal to the length of a longest oriented path plus one, which is a quite interesting result, as within undirected co-graphs even for bipartite graphs the path length can not be bounded by the chromatic number.

Further on oriented co-graphs an independent set of largest size can be computed in the same way as known for undirected co-graphs [6], as well as a tournament subdigraph of largest size and also a partition of the vertex set into a minimum number of tournaments can be computed by applying the method for independent set or oriented coloring problem on the reverse input graph.

It remains to consider the existence of an FPT-algorithm for OCN w.r.t. the parameter directed tree-width given in [24]. Since the directed tree-width of a digraph is always less or equal the undirected tree-width of the corresponding underlying undirected graph [24] the FPT-algorithm of Ganian [14] (see also Section 1) does not imply such a result.

In [14] Ganian has also shown that OCN is DET-hard for classes of oriented graphs such that the underlying undirected class has bounded rank-width. He used a reduction from isomorphism problem for tournaments, which has been shown to be DET-hard in [29]. The same reduction also works for several linear width parameters, since these can define the disjoint union of two arbitrarily large cliques. Thus OCN is DET-hard for classes of oriented graphs such that the underlying undirected class has linear NLC-width at most , linear clique-width at most , neighbourhood-width at most , or linear rank-width . Further OCN is DET-hard for classes of oriented graphs such that the underlying undirected class has NLC-width 1 or equivalently clique-width . The complexity of OCN on oriented graphs such that the underlying undirected class has linear NLC-width at most (equivalently neighbourhood-width ) or linear clique-width at most remains open, since these classes do not contain the disjoint union of two arbitrarily large cliques.

It also remains to generalize the shown results for oriented coloring on oriented graphs of bounded directed clique-width given in [8, 21]. By the existence of a monadic second order logic formula it follows that for every the problem is fixed parameter tractable w.r.t. the parameter directed clique-width.

For the more general problem OCN the existence of XP-algorithm or even FPT-algorithm w.r.t. 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 [21], the result of Ganian [14] (see also Section 1) does not imply a hardness result.

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