1 Introduction
During the last years classes of directed graphs have received a lot of attention [BJG18], since they are useful in multiple applications of directed networks. Meanwhile the class of directed cographs is used in applications in the field of genetics, see [NEMM18]. But the field of directed cographs is far from been as well studied as the undirected version, even though it has a similar useful structure. There are multiple subclasses of undirected cographs which were characterized successfully by different definitions. Meanwhile there are also corresponding subclasses of directed cographs as e.g. the class of oriented cographs, which has been analyzed by Lawler [Law76] and Boeckner [Boe18]. But there are many more interesting subclasses of directed cographs, that were mostly not characterized until now. Thus we consider directed versions of threshold graphs, simple cographs, trivially perfect graphs and weakly quasi threshold graphs. Furthermore, we take a look at the oriented versions of these classes and the related complement classes. All of these classes are hereditary, just like directed cographs, such that they can be characterized by a set of forbidden induced subdigraphs. We will even show a finite number of forbidden induced subdigraphs for the further introduced classes. This is for example very useful in the case of finding an efficient recognition algorithm for these classes.
Undirected cographs, i.e. complement reducible graphs, were developed independently by several authors, see [Ler71, Sum74] for example, while directed cographs were introduced 30 years later by Bechet et al. [BdGR97]. Due to their recursive structure there are in general hard problems, which can be solved efficiently on (directed) cographs. That makes this graph class particularly interesting.
This paper is organized as follows. After introducing some basic definitions we introduce undirected cographs in Section 3 and subclasses and recapitulate their relations and their characterizations by sets of forbidden subgraphs. In Section 4 we introduce directed and oriented cographs and summarize their properties. Subsequently, we focus on subclasses of directed cographs. We show definitions of seriesparallel partial order digraphs, directed trivially perfect graphs, directed weakly quasi threshold graphs, directed simple cographs, directed threshold graphs and the corresponding complementary and oriented versions of these classes. Some of the subclasses already exist, others are motivated by the related subclasses of undirected cographs given in Table 2. All of these subclasses have in common that they can be constructed recursively by several operations. Analogously to the undirected classes, we show how these multiple subclasses can be characterized by finite sets of minimal forbidden induced subdigraphs. We continue with an analysis of the relations of the several classes. Moreover, we analyze how they are related to the corresponding undirected classes. Finally in Section 5, we give conclusions including further research directions.
2 Preliminaries
2.1 Notations for Undirected Graphs
We work with finite undirected graphs , where is a finite set of vertices and is a finite set of edges. A graph is a subgraph of graph if and . If every edge of with both end vertices in is in , we say that is an induced subgraph of digraph and we write .
For some graph its complement digraph is defined by
For some graph class we define by .
For some graph some integer let be the disjoint union of copies of .
Special Undirected Graphs
As usual we denote by
a complete graph on vertices and by an edgeless graph on vertices, i.e. the complement graph of a complete graph on vertices. By
we denote a path on vertices. See Table 1 for examples.
2.2 Notations for Directed Graphs
A directed graph or digraph is a pair , where is a finite set of vertices and
is a finite set of ordered pairs of distinct
^{1}^{1}1Thus we do not consider directed graphs with loops. vertices called arcs. A vertex is outdominating (indominated) if it is adjacent to every other vertex in and is a source (a sink, respectively). A digraph is a subdigraph of digraph if and . If every arc of with both end vertices in is in , we say that is an induced subdigraph of digraph and we write .For some directed graph its complement digraph is defined by
and its converse digraph is defined by
For some digraph class we define by .
For some digraph some integer let be the disjoint union of copies of .
Notations for Directed Graphs
For a set of graphs we denote by free graphs the set of all graphs that do not contain a graph of as an induced subgraph.
Orientations
There are several ways to define a digraph from a undirected graph , see [BJG09]. If we replace every edge of by

one of the arcs and , we denote as an orientation of . Every digraph which can be obtained by an orientation of some undirected graph is called an oriented graph.

one or both of the arcs and , we denote as a biorientation of . Every digraph which can be obtained by a biorientation of some undirected graph is called a bioriented graph.

both arcs and , we denote as a complete biorientation of . Since in this case is well defined by we also denote it by . Every digraph which can be obtained by a complete biorientation of some undirected graph is called a complete bioriented graph.
For some given a digraph , we define its underlying undirected graph by ignoring the directions of the edges, i.e.
and for some class of digraphs , let
Special Directed Graphs
As usual we denote by
a bidirectional complete digraph on vertices and by an edgeless graph on vertices, i.e. the complement graph of a complete directed graph on vertices. By
a bidirectional complete bipartite digraph on vertices.
Special Oriented Graphs
By
we denote the oriented path on vertices. By
we denote the oriented cycle on vertices. By we denote a transitive tournament on vertices.^{2}^{2}2Please note that transitive tournaments on vertices are unique up to isomorphism, see [Gou12, Chapter 9] and [GRR18]. By
we note an oriented complete bipartite digraph on vertices.
An oriented forest (tree) is the orientation of a forest (tree). A digraph is an outtree (intree) if it is an oriented tree in which there is exactly one vertex of indegree (outdegree) zero.
2.3 Induced Subgraph Characterizations for Hereditary Classes
The following notations and results are given in [KL15, Chapter 2] for undirected graphs. These results also hold for directed graphs.
Classes of (di)graphs which are closed under taking induced sub(di)graphs are called hereditary. 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 .
Theorem 2.1 ([Kl15])
A class of (di)graphs is hereditary if and only if there is a set , such that .
A (di)graph is a minimal forbidden induced sub(di)graph for some hereditary class if does not belong to and every proper induced sub(di)graph of belongs to . For some hereditary (di)graph class we define as the set of all minimal forbidden induced sub(di)graphs for .
Theorem 2.2 ([Kl15])
For every hereditary class of (di)graphs it holds that . Set is unique and of minimal size.
Theorem 2.3 ([Kl15])
if and only if for every (di)graph there is a (di)graph such that is an induced sub(di)graph of .
Lemma 2.4 ([Kl15])
Let and be hereditary classes of (di)graphs. Then and .
Observation 2.5
Let be a digraph such that for a hereditary class of digraphs and there is some digraph such that all biorientations of are in , then .
Observation 2.6
Let be a digraph such that for some hereditary class of graphs , then for all and all biorientations of it holds that .
3 Undirected Cographs and Subclasses
In order to define cographs and subclasses we will use the following operations. Let and be two vertexdisjoint graphs.

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

The join of and , denoted by , is the graph with vertex set and edge set .
We also will recall forbidden induced subgraph characterizations for cographs and frequently analyzed subclasses. Therefore the graphs in Table 1 are very useful.
3.1 CoGraphs
Definition 3.1 (CoGraphs [Clsb81])
The class of cographs (short for complementreducible graphs) is defined recursively as follows.

Every graph on a single vertex , denoted by , is a cograph.

If and are cographs, then (a) and (b) are cographs.
The class of cographs is denoted by C.
3.2 Subclasses of CoGraphs
In Table 2 we summarize cographs and their wellknown subclasses. The given forbidden sets are known from the existing literature [CLSB81, Gol78, CH77, NP11, HMP11].
class  notation  operations  
cographs  C  
quasi threshold/trivially perfect graphs  TP  ,  
coquasi threshold/cotrivially perfect graphs  CTP  ,  
threshold graphs  T  , ,  
simple cographs  SC  , co,  
cosimple cographs  CSC  , ,  
weakly quasi threshold graphs  WQT  , co  
coweakly quasi threshold graphs  CWQT  ,  
edgeless graphs  
complete graphs  co  
disjoint union of two cliques  ,  
complete bipartite graphs  , co  
disjoint union of cliques  
join of stable sets  co 
3.3 Relations
In Figure 1 we compare the above graph classes to each other and show the hierarchy of the subclasses of cographs.
4 Directed CoGraphs and Subclasses
First we introduce operations in order to recall the definition of directed cographs from [BdGR97] and introduce some interesting subclasses. Let and be two vertexdisjoint directed graphs.^{3}^{3}3We use the same symbols for the disjoint union and join between undirected and directed graphs. Although the meaning becomes clear from the context we want to emphasize this fact.

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

The series composition of and , denoted by , is the digraph with vertex set and arc set .

The order composition of and , denoted by , is the digraph with vertex set and arc set .
Every graph structure which can be obtained by this operations, can be constructed by a tree or even a sequence, as we could do for undirected cographs and threshold graphs. These trees/sequences can be used for algorithmic properties of those graphs.
4.1 Directed CoGraphs
Definition 4.1 (Directed CoGraphs [BdGR97])
The class of directed cographs is recursively defined as follows.

Every digraph on a single vertex , denoted by , is a directed cograph.

If and are directed cographs, then (a) , (b) , and (c) are directed cographs.
The class of directed cographs is denoted by DC.
The recursive definition of directed and undirected cographs lead to the following observation.
Observation 4.2
For every directed cograph the underlying undirected graph is a cograph.
The reverse direction only holds under certain conditions, see Theorem 4.4.
Obviously for every directed cograph we can define a tree structure, denoted as binary dicotree. The leaves of the dicotree represent the vertices of the graph and the inner nodes of the dicotree correspond to the operations applied on the subexpressions defined by the subtrees. For every directed cograph one can construct a binary dicotree in linear time, see [CP06].
In [BJM14] it is shown that the weak linkage problem can be solved in polynomial time for directed cographs. 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 semi complete subdigraph, and the size of a largest complete subdigraph for every directed cograph in linear time. Also the hamiltonian path, hamiltonian cycle, regular subdigraph, and directed cut problem are polynomial on directed cographs [Gur17]. Calculs of directed cographs were also considered in connection with pomset logic in [Ret99]. Further the directed pathwidth, directed treewidth, directed feedback vertex set number, cycle rank, DAGdepth and DAGwidth can be computed in linear time for directed cographs [GKR19a].
Lemma 4.3 ([GR18a])
Let be some digraph, then the following properties hold.

Digraph is a directed cograph if and only if digraph is a directed cograph.

Digraph is a directed cograph if and only if digraph is a directed cograph.
It further hold the following properties for directed cographs:
Theorem 4.4
Let be a digraph. The following properties are equivalent:

is a directed cograph.

.

and .

and is a cograph.

has directed NLCwidth .

has directed cliquewidth at most and .
For subclasses of directed cographs, which will be defined in the following subsections, some more forbidden subdigraphs are needed. Those are defined in Tables 4, 5, and 6.






Observation 4.5
It holds:

.

.
For directed cographs Observation 4.5 leads to the next result.
Proposition 4.6
.
4.2 Oriented CoGraphs
Beside directed cographs and their subclasses we also will restrict the these classes to oriented graphs by omitting the series operation.
Definition 4.7 (Oriented CoGraphs)
The class of oriented cographs is recursively defined as follows.

Every digraph on a single vertex , denoted by , is an oriented cograph.

If and are oriented cographs, then (a) and (b) are oriented cographs.
The class of oriented cographs is denoted by OC.
The recursive definition of oriented and undirected cographs lead to the following observation.
Observation 4.8
For every oriented cograph the underlying undirected graph is a cograph.
The reverse direction only holds under certain conditions, see Theorem 4.10. The class of oriented cographs was already analyzed by Lawler in [Law76] and [CLSB81, 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 . For oriented cographs the oriented chromatic number and also the graph isomorphism problem can be solved in linear time [GKR19b].
Lemma 4.9
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. ∎
The class OC can also be defined by forbidden subdigraphs.
Theorem 4.10
Let be a digraph. The following properties are equivalent:

is an oriented cograph

.

and .

and is a cograph.

directed has NLCwidth and .

has directed cliquewidth at most and .

is transitive and .
Proof.
If is an oriented cograph, then is a directed cograph and by Theorem 4.4 it holds that . Further because of the missing series composition. This leads to .
If then and is a directed cograph. Since there is no series operation in any construction of which implies that is an oriented cograph.
Since .
By Lemma 4.9 we know that is transitive. If is transitive, then .
and By Theorem 4.4.
Observation 4.11
Every oriented cograph is a DAG.
Theorem 4.12 ([Clsb81])
A graph is a cograph if and only if there exists an orientation of such that is an oriented cograph.
4.3 Seriesparallel partial order digraphs
We recall the definitions of from [BJG18] which are based on [VTL82]. A seriesparallel partial order is a partially ordered set that is constructed by the series composition and the parallel composition operation starting with a single element.

Let and be two disjoint seriesparallel partial orders, then distinct elements of a series composition^{4}^{4}4Note that the series composition in this case corresponds to the order composition in the definition of directed cographs. have the same order they have in or . Respectively, this holds if both of them are from the same set, and , if and .

Two elements of a parallel composition are comparable if and only if both of them are in or both in , while they keep their corresponding order.
Definition 4.13 (Seriesparallel partial order digraphs)
A seriesparallel partial order digraph is a digraph, where is a seriesparallel partial order and if and only if and .
The class of seriesparallel partial order digraphs is denoted by SPO.
For a digraph an edge is symmetric, if . Thus each bidirectional arc is symmetric. Further, an edge is asymmetric, if it is not symmetric, i.e. each edge with only one direction. We define the symmetric part of as , which is the spanning subdigraph of that contains exactly the symmetric arcs of . Analogously we define the asymmetric part of as , which is the spanning subdigraph with only asymmetric edges.
Moreover, Bechet et al. showed in [BdGR97] the following property of directed cographs.
Lemma 4.14 ([BdGR97])
For every directed cograph it holds that the asymmetric part of is a seriesparallel partial order digraph and for the symmetric part the underlying undirected graph a cograph.
The class of seriesparallel partial ordered digraph is equal to the class of oriented cographs, since they have exactly the same recursive structure. Thus this lemma is easy to prove with the following idea. Let be a directed cograph and its corresponding dicotree.

symmetric part: Exchange each order composition with a directed union composition. Since there are no more oriented arcs left, this tree represents a cograph.

asymmetric part: Exchange each series composition with a directed union composition. Since there are no more bidirectional edges left, this tree represents an oriented cograph, e.g. a seriesparallel partial order digraph.
4.4 Directed trivially perfect graphs
Definition 4.15 (Directed trivially perfect graphs)
The class of directed trivially perfect graphs is recursively defined as follows.

Every digraph on a single vertex , denoted by , is a directed trivially perfect graph.

If and are directed trivially perfect graphs, then is a directed trivially perfect graph.

If is a directed trivially perfect graph, then (a) , (b) , and (c) are directed trivially perfect graphs.
The class of directed trivially perfect graphs is denoted by DTP.
The recursive definition of directed and undirected trivially perfect graphs lead to the following observation.
Observation 4.16
For every directed trivially perfect graph the underlying undirected graph is a trivially perfect graph.
The reverse direction only holds under certain conditions, see Theorem 4.18.
Lemma 4.17 ([Grr18])
For every digraph the following statements are equivalent.

is a transitive tournament.

is an acyclic tournament.

and is a tournament.

can be constructed from the onevertex graph by repeatedly adding an outdominating vertex.

can be constructed from the onevertex graph by repeatedly adding an indominated vertex.
The class DTP can also be defined by forbidden induced subdigraphs. It holds that
Theorem 4.18
Let be a digraph. The following properties are equivalent:

is a directed trivially perfect graph.

.

and .

and is a trivially perfect graph.
Proof.
The given forbidden digraphs are not directed trivially perfect graphs and the set of directed trivially perfect graphs is closed under taking induced subdigraphs.
Since digraph is a directed cograph by [CP06] and thus has a construction using disjoint union, series composition, and order composition.
Since we know that for every series combination between two graphs on at least two vertices at least one of the graphs is bidirectional complete. Such a subgraph can be inserted by a number of feasible operations for directed trivially perfect graphs.
Since we know that for every order combination between two graphs on at least two vertices at least one of the graphs is a tournament. Since by Lemma 4.17 we even know that at least one of the graphs is a transitive tournament. Such a graph can be defined by a sequence of outdominating or indominating vertices (Lemma 4.17) which are also feasible operations for directed trivially perfect graphs.
For directed trivially perfect graphs Observation 4.5 leads to the next result.
Proposition 4.19
This motivates us to consider the class of edge complements of directed trivially perfect graphs.
Definition 4.20 (Directed cotrivially perfect graphs)
The class of directed cotrivially perfect graphs is recursively defined as follows.

Every digraph on a single vertex , denoted by , is a directed cotrivially perfect graph.

If and are directed cotrivially perfect graphs, then is a directed trivially perfect graph.

If is a directed cotrivially perfect graph, then (a) , (b) , and (c) are directed cotrivially perfect graphs.
The class of directed cotrivially perfect graphs is denoted by DCTP.
Theorem 4.18 and Lemma 2.4 lead to the following characterization for directed cotrivially perfect graphs.
Theorem 4.21
Let be a digraph. The following properties are equivalent:

is a directed cotrivially perfect graph.

.

and .

and is a cotrivially perfect graph.
4.5 Oriented trivially perfect graphs
Definition 4.22 (Oriented trivially perfect graphs)
The class of oriented trivially perfect graphs is recursively defined as follows.

Every digraph on a single vertex , denoted by , is an oriented trivially perfect graph.

If and are oriented trivially perfect graphs, then is an oriented trivially perfect graph.

If is an oriented trivially perfect graph, then (a) and (b) are oriented trivially perfect graphs.
The class of oriented trivially perfect graphs is denoted by OTP.
The recursive definition of oriented and undirected trivially perfect graphs lead to the following observation.
Observation 4.23
For every oriented trivially perfect graph the underlying undirected graph is a trivially perfect graph.
Similar as for oriented cographs we obtain a definition of OTP by forbidden induced subdigraphs.
Theorem 4.24
Let be a digraph. The following properties are equivalent:

is an oriented trivially perfect graph.

.

and .

and is a trivially perfect graph.

is transitive and .
Proof.
If is an oriented trivially perfect graph, then is a directed trivially perfect graph and . Further because of the missing series composition. This leads to .
If , then and is a directed trivially perfect graph. Since there is no series operation in any construction of which implies that is an oriented trivially perfect graph.
Since .
By Lemma 4.9 we know that is transitive. If is transitive, then has no induced . ∎
Theorem 4.25
A graph is a trivially perfect graph if and only if there exists an orientation of such that is an oriented trivially perfect graph.
Proof.
Let be a trivially perfect graph. Then is also a comparability graph, which implies that has a transitive orientation . Since it follows that . Further by definition . By Theorem 4.24 we know that is an oriented trivially perfect graph.
For the reverse direction let be an oriented trivially perfect graph. Then by Theorem 4.24 it holds that . Since is the only transitive orientation of and is the only transitive orientation of it holds that . Thus is a trivially perfect graph. ∎
Observation 4.26
If then the underlying undirected graph of the symmetric part of is trivially perfect and the asymmetric part of is an oriented trivially perfect digraph.
This holds since the asymmetric part is exactly build with the same rules like trivially perfect graphs and the asymmetric part with the rules of OTP.
Definition 4.27 (Oriented cotrivially perfect graphs)
The class of oriented cotrivially perfect graphs is recursively defined as follows.

Every digraph on a single vertex , denoted by , is an oriented cotrivially perfect graph.

If is an oriented cotrivially perfect graph, then (a) , (b) , and (c) are oriented cotrivially perfect graphs.
The class of oriented cotrivially perfect graphs is denoted by OCTP.
Restricting the operations of directed cotrivially perfect graphs to oriented graphs leads to the same operations as the class of orientated threshold graphs, which will be considered in Section 4.15.
4.6 Directed Weakly Quasi Threshold Graphs
Definition 4.28 (Directed weakly quasi threshold graphs)
The class of directed weakly quasi threshold graphs is recursively defined as follows.

Every edgeless digraph is a directed weakly quasi threshold graph.

If and are directed weakly quasi threshold graphs, then is a directed weakly quasi threshold graph.

If is a directed weakly quasi threshold graph and is an edgeless digraph, then (a) , (b) , and (c) are directed weakly quasi threshold graphs.
The class of directed weakly quasi threshold graphs is denoted by DWQT.
Observation 4.29
If is a directed weakly quasi threshold graph, is weakly quasi threshold graph.
Theorem 4.30
Let be a digraph. The following properties are equivalent:

is a directed weakly quasi threshold graph.

^{5}^{5}5Note that and ..

and .

and is a weakly quasi threshold graph.
Proof.
The given forbidden digraphs are not directed weakly quasi threshold graphs and the set of directed weakly quasi threshold graphs is hereditary.
Let be a digraph without induced . Since there are no induced , it holds that . Thus, is constructed by the disjoint union, the series and the order composition. and can only be build by a series composition of two graphs and , where . We note that are contained in every directed cograph containing more vertices, that is not a sequence of length at least one of series compositions of independent sets. Consequently, there are no bigger forbidden induced subdigraphs that emerged through a series operation, such that the and characterize exactly the allowed series compositions in DWQT.
The and can only be build by an order composition of two graphs and , where . We note that are contained in every directed cograph containing more vertices, that is not a sequence of length at least one of order compositions of independent sets. Consequently, there are no forbidden induced subdigraphs containing more vertices that emerged through an order operation, such that the and characterize exactly the allowed order compositions in DWQT. Finally, by excluding these digraphs, we end up in the Definition 4.28 for DWQT, such that .
By Observation 4.29.
By Observation 2.6.
Since . ∎
Comments
There are no comments yet.