On Characterizations for Subclasses of Directed Co-Graphs

TL;DR

This paper characterizes various subclasses of directed co-graphs using forbidden subdigraphs and explores their relationships, extending known undirected graph classes to directed contexts.

Contribution

It introduces characterizations of multiple directed co-graph subclasses via forbidden subdigraphs, expanding the understanding of their structure and interrelations.

Findings

Characterizations of subclasses by forbidden subdigraphs

Relations between different directed co-graph classes

Extension of undirected subclasses to directed co-graphs

Abstract

Undirected co-graphs are those graphs which can be generated from the single vertex graph by disjoint union and join operations. Co-graphs are exactly the P_4-free graphs (where P_4 denotes the path on 4 vertices). Co-graphs itself and several subclasses haven been intensively studied. Among these are trivially perfect graphs, threshold graphs, weakly quasi threshold graphs, and simple co-graphs. Directed co-graphs are precisely those digraphs which can be defined from the single vertex graph by applying the disjoint union, order composition, and series composition. By omitting the series composition we obtain the subclass of oriented co-graphs which has been analyzed by Lawler in the 1970s and the restriction to linear expressions was recently studied by Boeckner. There are only a few versions of subclasses of directed co-graphs until now. By transmitting the restrictions of undirected subclasses to the directed classes, we define the corresponding subclasses for directed co-graphs. We consider directed and oriented versions of threshold graphs, simple co-graphs, co-simple co-graphs, trivially perfect graphs, co-trivially perfect graphs, weakly quasi threshold graphs and co-weakly quasi threshold graphs. For all these classes we provide characterizations by finite sets of minimal forbidden induced subdigraphs. Further we analyze relations between these graph classes.

Tables

Table 1. Table 1: Special graphs

$P_4$		$C_4$		$2K_2$		$2P_3$

Table 2. Table 2: Overview on subclasses of co-graphs. By G 1 subscript 𝐺 1 G_{1} and G 2 subscript 𝐺 2 G_{2} we denote graphs of the class X 𝑋 X , by I 𝐼 I we denote an edgeless graph and by K 𝐾 K we denote a complete graph. Classes and complement classes are listed between two horizontal lines. That is, only co-graphs and threshold graphs are closed under taking edge complementation.

class $X$	notation	operations			$\text{Forb}\left(X\right)$
co-graphs	C	$\bullet$	$G_1\oplus G_2$	$G_1\otimes G_2$	$P_4$
quasi threshold/trivially perfect graphs	TP	$\bullet$	$G_1\oplus G_2$	$G_1\otimes \bullet$	$P_4$, $C_4$
co-quasi threshold/co-trivially perfect graphs	CTP	$\bullet$	$G_1\oplus \bullet$	$G_1\otimes G_2$	$P_4$, $2K_2$
threshold graphs	T	$\bullet$	$G_1\oplus \bullet$	$G_1\otimes \bullet$	$P_4$, $C_4$, $2K_2$
simple co-graphs	SC	$\bullet$	$G_1\oplus I$	$G_1\otimes I$	$P_4$, co-$2P_3$, $2K_2$
co-simple co-graphs	CSC	$\bullet$	$G_1\oplus K$	$G_1\otimes K$	$P_4$, $2P_3$, $C_4$
weakly quasi threshold graphs	WQT	$I$	$G_1\oplus G_2$	$G_1\otimes I$	$P_4$, co-$2P_3$
co-weakly quasi threshold graphs	CWQT	$K$	$G_1\oplus K$	$G_1\otimes G_2$	$P_4$, $2P_3$
edgeless graphs		$\bullet$	$G_1\oplus \bullet$		$P_2$
complete graphs		$\bullet$		$G_1\otimes \bullet$	co-$P_2$
disjoint union of two cliques			$K\oplus K$		$I_3$ ,$P_3$
complete bipartite graphs				$I\otimes I$	$K_3$, co-$P_3$
disjoint union of cliques		$K$	$G_1\oplus K$		$P_3$
join of stable sets		$I$		$G_1\otimes I$	co-$P_3$

Table 3. Table 3: The eight forbidden induced subdigraphs for directed co-graphs.

$D_1$		$D_2$		$D_3$		$D_4$
$D_5$		$D_6$		$D_7$		$D_8$

Table 4. Table 4: Forbidden induced subdigraphs for subclasses of directed co-graphs.

$D_9$		$D_{10}$		$D_{11}$

Table 5. Table 5: Forbidden induced subdigraphs for subclasses of directed co-graphs.

$D_{12}$		$D_{13}$		$D_{14}$		$D_{15}$

Table 6. Table 6: Forbidden induced subdigraphs for subclasses of directed co-graphs.

$\overleftrightarrow{K_2}$		$2\overrightarrow{P_2}=\text{co-}{D}_{11}$		$\text{co-}{D}_{10}$		$\text{co-}{D}_9=\text{co-}{D}_9$

Table 7. Table 7: Digraphs to build the forbidden subdigraphs of DWQT .

Table 8. Table 8: Constructions of the forbidden subdigraphs of DWQT .

$Q_1=Y_2\otimes Y_2$	$Q_2=Y_1\oslash Y_1$	$Q_3=Y_3\otimes Y_3$	$Q_4=Y_2\otimes Y_3$
$Q_5=Y_1\oslash Y_4$	$Q_6=Y_4\oslash Y_1$	$Q_7=Y_4\oslash Y_4$

Table 9. Table 14: Overview on subclasses of directed co-graphs. By G 1 subscript 𝐺 1 G_{1} and G 2 subscript 𝐺 2 G_{2} we denote graphs of the class X 𝑋 X , by I 𝐼 I we denote an edgeless graph, by K 𝐾 K we denote a bidirectional complete digraph, and by T 𝑇 T we denote a transitive tournament. Classes and complement classes are listed between two horizontal lines. That is, only directed co-graphs and directed threshold graphs are closed under edge complementation.

class $X$	notation	operations					$\text{Forb}\left(X\right)$
directed co-graphs	DC	$\bullet$	$G_1\oplus G_2$	$G_1\oslash G_2$		$G_1\otimes G_2$	$D_1,\dots ,D_8$
oriented co-graphs	OC	$\bullet$	$G_1\oplus G_2$	$G_1\oslash G_2$			$D_1,D_5,D_8,\overleftrightarrow{K_2}$
directed trivially perfect	DTP	$\bullet$	$G_1\oplus G_2$	$G_1\oslash \bullet$	$\bullet \oslash G_2$	$G_1\otimes \bullet$	$D_1,\dots ,D_{15}$
oriented trivially perfect	OTP	$\bullet$	$G_1\oplus G_2$	$G_1\oslash \bullet$	$\bullet \oslash G_2$		$D_1,D_5,D_8,\overleftrightarrow{K_2},D_{12}$
directed co-trivially perfect	DCTP	$\bullet$	$G_1\oplus \bullet$	$G_1\oslash \bullet$	$\bullet \oslash G_2$	$G_1\otimes G_2$	$D_1,\dots ,D_8$, $D_{12},\dots ,D_{15}$
							$\text{co-}{D}_{11},\text{co-}{D}_{10},\text{co-}{D}_9$
oriented co-trivially	OCTP	$\bullet$	$G_1\oplus \bullet$	$G_1\oslash \bullet$	$\bullet \oslash G_2$		$D_1,D_5,D_8,D_{12},$
perfect $\ast$							$\text{co-}{D}_{11},\overleftrightarrow{K_2}$
directed weakly	DWQT	$I$	$G_1\oplus G_2$	$G_1\oslash I$	$I\oslash G_2$	$G_1\otimes I$	$D_1,\dots ,D_8,Q_1,\dots ,Q_7$
quasi threshold
oriented weakly	OWQT	$I$	$G_1\oplus G_2$	$G_1\oslash I$	$I\oslash G_2$		$D_1,D_5,D_8,\overleftrightarrow{K_2},Q_7$
quasi threshold
directed co-weakly	DCWQT	$K$	$G_1\oplus K$	$G_1\oslash K$	$K\oslash G_2$	$G_1\otimes G_2$	$D_1,\dots ,D_8,\text{co-}{Q}_1,\dots ,\text{co-}{Q}_7$
quasi threshold
oriented co-weakly	OCWQT	$T$	$G_1\oplus T$	$G_1\oslash T$	$T\oslash G_2$		$D_1,D_5,D_8,\overleftrightarrow{K_2},$
quasi threshold							$D_{12},D_{21},D_{22},D_{23}$
directed simple co-graphs	DSC	$\bullet$	$G_1\oplus I$	$G_1\oslash I$	$I\oslash G_2$	$G_1\otimes I$	$D_1,\dots ,D_8,Q_1,\dots ,Q_7,$
							$\text{co-}{D}_9,\text{co-}{D}_{10},\text{co-}{D}_{11}$
oriented simple co-graphs	OSC	$\bullet$	$G_1\oplus I$	$G_1\oslash I$	$I\oslash G_2$		$D_1,D_5,D_8,\overleftrightarrow{K_2},Q_7,\text{co-}{D}_{11}$
directed co-simple co-graphs	DCSC	$\bullet$	$G_1\oplus K$	$G_1\oslash K$	$K\oslash G_2$	$G_1\otimes K$	$D_1,\dots ,D_8,Q_1,$
							$\text{co-}{Q}_1,\dots ,\text{co-}{Q}_7,D_9,D_{10}$
oriented co-simple co-graphs	OCSC	$\bullet$	$G_1\oplus T$	$G_1\oslash T$	$T\oslash G_2$		$D_1,D_5,D_8,\overleftrightarrow{K_2},$
							$D_{12},D_{21},D_{22},D_{23}$
directed threshold graphs	DT	$\bullet$	$G_1\oplus \bullet$	$G_1\oslash \bullet$	$\bullet \oslash G_2$	$G_1\otimes \bullet$	$D_1,\dots ,D_{15}$
							$\text{co-}{D}_{11},\text{co-}{D}_{10},\text{co-}{D}_9$
oriented threshold graphs $\ast$	OT	$\bullet$	$G_1\oplus \bullet$	$G_1\oslash \bullet$	$\bullet \oslash G_2$		$D_1,D_5,D_8,\overleftrightarrow{K_2},D_{12},\text{co-}{D}_{11}$
threshold digraphs	TD						$D_5$, 2-switch
Ferres digraphs	FD						$D_1,\overleftrightarrow{K_2}$, 2-switch
edgeless digraphs		$\bullet$	$G_1\oplus \bullet$				$\overrightarrow{P_2}$, $\overleftrightarrow{K_2}$
bidirectional		$\bullet$				$G_1\otimes \bullet$	$\overrightarrow{P_2}$, $\text{co-}\overleftrightarrow{K_2}$
complete digraphs
transitive tournaments	TT	$\bullet$		$G_1\oslash \bullet$			$2\overleftrightarrow{K_1},\overleftrightarrow{K_2},D_5$
bidirectional complete						$I\otimes I$	co-$\overleftrightarrow{P_3},\overrightarrow{P_2},\overleftrightarrow{K_3}$
bipartite digraphs
disjoint union of			$K\oplus K$				$\overleftrightarrow{P_3},\overrightarrow{P_2},\overleftrightarrow{I_3}$
2 bidirectional complete
series composition of		$I$				$G_1\otimes I$	co-$\overleftrightarrow{P_3},\overrightarrow{P_2}$
stable sets
disjoint union of		$K$	$G_1\oplus K$				$\overleftrightarrow{P_3},\overrightarrow{P_2}$
bidirectional complete

Equations

$$\text{co-}G=(V,\{\{u,v\}|\{u,v\}\notin E,u,v\in V,u\ne v\}).$$

$$K_n=(\{v_1,\dots ,v_n\},\{\{v_i,v_j\}|1\le i<j\le n\}),$$

$$P_n=(\{v_1,\dots ,v_n\},\{\{v_1,v_2\},\dots ,\{v_{n-1},v_n\}\})$$

$$\text{co-}G=(V,\{(u,v)|(u,v)\notin E,u,v\in V,u\ne v\})$$

$$G^c=(V,\{(u,v)|(v,u)\in E,u,v\in V,u\ne v\}).$$

$$un(D)=(V,\{\{u,v\}|(u,v)\in A,u,v\in V\}).$$

$$un(X)=\{un(D)|D\in X\}.$$

$$\overleftrightarrow{K_n}=(\{v_1,\dots ,v_n\},\{(v_i,v_j)|1\le i\ne j\le n\}),$$

$$\overleftrightarrow{K_{n,m}}=(\{v_1,\dots ,v_n,w_1,\dots ,w_m\},\{(v_i,w_j),(w_j,v_i)|1\le i\le n,1\le j\le m\}),$$

$$\overrightarrow{P_n}=(\{v_1,\dots ,v_n\},\{(v_1,v_2),\dots ,(v_{n-1},v_n)\})$$

$$\overrightarrow{C_n}=(\{v_1,\dots ,v_n\},\{(v_1,v_2),\dots ,(v_{n-1},v_n),(v_n,v_1)\})$$

$$\overrightarrow{K_{n,m}}=(\{v_1,\dots ,v_n,w_1,\dots ,w_m\},\{(v_i,w_j)|1\le i\le n,1\le j\le m\}),$$

Full text

Directed co-graphs are precisely those digraphs which can be defined from the single vertex graph by applying the disjoint union, order composition, and series composition. By omitting the series composition we obtain the subclass of oriented co-graphs which has been analyzed by Lawler in the 1970s and the restriction to linear expressions was recently studied by Boeckner. There are only a few versions of subclasses of directed co-graphs until now. By transmitting the restrictions of undirected subclasses to the directed classes, we define the corresponding subclasses for directed co-graphs. We consider directed and oriented versions of threshold graphs, simple co-graphs, co-simple co-graphs, trivially perfect graphs, co-trivially perfect graphs, weakly quasi threshold graphs and co-weakly quasi threshold graphs. For all these classes we provide characterizations by finite sets of minimal forbidden induced subdigraphs. Further we analyze relations between these graph classes.

Keywords: co-graphs; directed co-graphs; directed threshold graphs; digraphs; threshold graphs; forbidden induced subdigraph characterizations

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 co-graphs is used in applications in the field of genetics, see [NEMM*+*18]. But the field of directed co-graphs 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 co-graphs which were characterized successfully by different definitions. Meanwhile there are also corresponding subclasses of directed co-graphs as e.g. the class of oriented co-graphs, which has been analyzed by Lawler [Law76] and Boeckner [Boe18]. But there are many more interesting subclasses of directed co-graphs, that were mostly not characterized until now. Thus we consider directed versions of threshold graphs, simple co-graphs, 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 co-graphs, 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 a polynomial recognition algorithm for these classes.

Undirected co-graphs, i.e. complement reducible graphs, appeared independently by several authors, see [Ler71, Sum74] for example, while directed co-graphs were introduced 30 years later by Bechet et al. [BdGR97]. Due to their recursive structure there are problems, that are hard in general, which can be solved efficiently on (directed) co-graphs, see [BM93, CLSB81, CPS84, LOP95, GHK*+*20b] and [BJM14, Gur17, GKR19b, GKR19c, GR18b, GHK*+*20a, GKL20]. That makes both graph classes particularly interesting. This paper can be seen a basis for further research on possible algorithms that are efficient for the structures of the presented graph classes.

This paper is organized as follows. After introducing some basic definitions we introduce undirected co-graphs 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 co-graphs and summarize their properties. Subsequently, we focus on subclasses of directed co-graphs. We show definitions of series-parallel partial order digraphs, directed trivially perfect graphs, directed weakly quasi threshold graphs, directed simple co-graphs, 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 co-graphs 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 $G=(V,E)$, where $V$ is a finite set of vertices and $E\subseteq\{\{u,v\}\mid u,v\in V,~{}u\not=v\}$ is a finite set of edges. A graph $G^{\prime}=(V^{\prime},E^{\prime})$ is a subgraph of graph $G=(V,E)$ if $V^{\prime}\subseteq V$ and $E^{\prime}\subseteq E$. If every edge of $E$ with both end vertices in $V^{\prime}$ is in $E^{\prime}$, we say that $G^{\prime}$ is an induced subgraph of digraph $G$ and we write $G^{\prime}=G[V^{\prime}]$.

For some graph $G=(V,E)$ its complement digraph is defined by

[TABLE]

For some graph class $X$ we define by $\text{co-}X=\{\text{co-}G~{}|~{}G\in X\}$.

For some graph $G$ some integer $d$ let $dG$ be the disjoint union of $k$ copies of $G$.

Special Undirected Graphs

As usual we denote by

[TABLE]

$n\geq 1$ a complete graph on $n$ vertices and by $I_{n}$ an edgeless graph on $n$ vertices, i.e. the complement graph of a complete graph on $n$ vertices. By

[TABLE]

we denote a path on $n$ vertices. See Table 1 for examples.

2.2 Notations for Directed Graphs

A directed graph or digraph is a pair $G=(V,E)$, where $V$ is a finite set of vertices and $E\subseteq\{(u,v)\mid u,v\in V,~{}u\not=v\}$ is a finite set of ordered pairs of distinct111Thus, we do not consider directed graphs with loops or multiple edges. vertices called arcs. A vertex $v\in V$ is out-dominating (in-dominated) if it is adjacent to every other vertex in $V$ and is a source (a sink, respectively). A digraph $G^{\prime}=(V^{\prime},E^{\prime})$ is a subdigraph of digraph $G=(V,E)$ if $V^{\prime}\subseteq V$ and $E^{\prime}\subseteq E$. If every arc of $E$ with both end vertices in $V^{\prime}$ is in $E^{\prime}$, we say that $G^{\prime}$ is an induced subdigraph of digraph $G$ and we write $G^{\prime}=G[V^{\prime}]$.

For a directed graph $G=(V,E)$ its complement digraph is defined by

[TABLE]

and its converse digraph is defined by

[TABLE]

For a digraph class $X$ we define by $\text{co-}X=\{\text{co-}G~{}|~{}G\in X\}$. For a digraph $G$ and an integer $d$ let $dG$ be the disjoint union of $k$ copies of $G$.

Orientations

There are several ways to define a digraph $D=(V,A)$ from an undirected graph $G=(V,E)$, see [BJG09]. If we replace every edge $\{u,v\}$ of $G$ by

• •

one of the arcs $(u,v)$ and $(v,u)$, we denote $D$ as an orientation of $G$. Every digraph $D$ that can be obtained by an orientation of an undirected graph $G$ is called an oriented graph.

• •

one or both of the arcs $(u,v)$ and $(v,u)$, we denote $D$ as a biorientation of $G$. A digraph $D$ that we get by a biorientation of an undirected graph $G$ is called a bioriented graph.

• •

both arcs $(u,v)$ and $(v,u)$, we denote $D$ as a complete biorientation of $G$. Since in this case $D$ is well defined by $G$ we also denote it by $\overleftrightarrow{G}$. A digraph $D$ that we obtain by a complete biorientation of an undirected graph $G$ is called a complete bioriented graph.

For a given digraph $D=(V,A)$ we define its underlying undirected graph by ignoring the directions of the edges and deleting multiple edges, i.e.

[TABLE]

and for some class of digraphs $X$, let

[TABLE]

Special Directed Graphs

As usual we denote by

[TABLE]

$n\geq 1$ a bidirectional complete digraph on $n$ vertices and by $\overleftrightarrow{I_{n}}$ an edgeless graph with $n$ vertices, i.e. the complement graph of a complete directed graph on $n$ vertices. By

[TABLE]

$n,m\geq 1$ a bidirectional complete bipartite digraph with $n+m$ vertices. A directed clique is a bidirectional complete digraph, such that $K=(V,E)$ with $E=\{(u,v)\mid\forall u,v\in V,u\neq v\}$ holds.

Special Oriented Graphs

By

[TABLE]

we denote the oriented path on $n$ vertices. By

[TABLE]

we denote the oriented cycle on $n$ vertices. By $T_{n}$ we denote a transitive tournament on $n$ vertices.222Please note that transitive tournaments on $n$ vertices are unique up to isomorphism, see [Gou12, Chapter 9], [GRR18] and Lemma 4.17. By

[TABLE]

$n,m\geq 1$ we denote an oriented complete bipartite digraph on $n+m$ vertices.

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 $F$ we define $\text{Free}(F)$ as the set of all (di)graphs $G$ such that no induced sub(di)graph of $G$ is isomorphic to a member of $F$.

Theorem 2.1** **([KL15])

A class of (di)graphs $X$ is hereditary if and only if there is a set $F$, such that $\text{Free}(F)=X$.

A (di)graph $G$ is a minimal forbidden induced sub(di)graph for some hereditary class $X$ if $G$ does not belong to $X$ and every proper induced sub(di)graph of $G$ belongs to $X$. For some hereditary (di)graph class $X$ we define $\text{Forb}(X)$ as the set of all minimal forbidden induced sub(di)graphs for $X$.

Theorem 2.2** **([KL15])

For every hereditary class of (di)graphs $X$ it holds that $X=\text{Free}(\text{Forb}(X))$. Set $\text{Forb}(X)$ is unique and of minimal size.

Theorem 2.3** **([KL15])

$\text{Free}(F_{1})\subseteq\text{Free}(F_{2})$* if and only if for every (di)graph $G\in F_{2}$ there is a (di)graph $H\in F_{1}$ such that $H$ is an induced sub(di)graph of $G$.*

Lemma 2.4** **([KL15])

Let $X=\text{Free}(F_{1})$ and $Y=\text{Free}(F_{2})$ be hereditary classes of (di)graphs. Then $X\cap Y=\text{Free}(F_{1}\cup F_{2})$ and $\text{co-}X=\text{Free}(\text{co-}F_{1})$.

Observation 2.5

Let $G$ be a digraph such that $G\in\text{Free}(X)$ for a hereditary class of digraphs $\text{Free}(X)$ and there is some digraph $X^{*}\in X$ such that all biorientations of ${\it un}(X^{*})$ are in $\text{Free}(X)$, then ${\it un}(G)\in\text{Free}({\it un}(X^{*}))$.

Observation 2.6

Let $G$ be a digraph such that ${\it un}(G)\in\text{Free}(X)$ for some hereditary class of graphs $\text{Free}(X)$, then for all $X^{*}\in X$ and all biorientations $b(X^{*})$ of $X^{*}$ it holds that $G\in\text{Free}(b(X^{*}))$.

3 Undirected Co-graphs and Subclasses

In order to define co-graphs and subclasses we will use the following operations. Let $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$ be two vertex-disjoint graphs.

• •

The disjoint union of $G_{1}$ and $G_{2}$, denoted by $G_{1}\oplus G_{2}$, is the graph with vertex set $V_{1}\cup V_{2}$ and edge set $E_{1}\cup E_{2}$.

• •

The join of $G_{1}$ and $G_{2}$, denoted by $G_{1}\otimes G_{2}$, is the graph with vertex set $V_{1}\cup V_{2}$ and edge set $E_{1}\cup E_{2}\cup\{\{v_{1},v_{2}\}~{}|~{}v_{1}\in V_{1},v_{2}\in V_{2}\}$.

We also will recall forbidden induced subgraph characterizations for co-graphs and frequently analyzed subclasses. Therefore the graphs in Table 1 are very useful.

3.1 Co-Graphs

Definition 3.1** **(Co-Graphs [CLSB81])

The class of co-graphs (short for complement-reducible graphs) is defined recursively as follows.

1. (i)

Every graph on a single vertex $(\{v\},\emptyset)$, denoted by $\bullet$, is a co-graph. 2. (ii)

*If $G_{1}$ and $G_{2}$ are co-graphs, then *

(a) $G_{1}\oplus G_{2}$*and * (b) $G_{1}\otimes G_{2}$*are co-graphs. * **

The class of co-graphs is denoted by C.

Using the recursive structure of co-graphs many problems can be solved in linear time, e.g. see [CLSB81]. Further the recursive structure allows to compute the path-width and tree-width of co-graphs in linear time [BM93].

3.2 Subclasses of Co-Graphs

In Table 2 we summarize co-graphs and their well-known subclasses. The given forbidden sets are known from the existing literature [CLSB81, Gol78, CH77, NP11, HMP11].

3.3 Relations

In Figure 1 we compare the above graph classes to each other and show the hierarchy of the subclasses of co-graphs.

4 Directed Co-Graphs and Subclasses

First we introduce operations in order to recall the definition of directed co-graphs from [BdGR97] and introduce some interesting subclasses. Let $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$ be two vertex-disjoint directed graphs.333We 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 $G_{1}$ and $G_{2}$, denoted by $G_{1}\oplus G_{2}$, is the digraph with vertex set $V_{1}\cup V_{2}$ and arc set $E_{1}\cup E_{2}$.

• •

The series composition of $G_{1}$ and $G_{2}$, denoted by $G_{1}\otimes G_{2}$, is the digraph with vertex set $V_{1}\cup V_{2}$ and arc set $E_{1}\cup E_{2}\cup\{(u,v),(v,u)~{}|~{}u\in V_{1},v\in V_{2}\}$.

• •

The order composition of $G_{1}$ and $G_{2}$, denoted by $G_{1}\oslash G_{2}$, is the digraph with vertex set $V_{1}\cup V_{2}$ and arc set $E_{1}\cup E_{2}\cup\{(u,v)~{}|~{}u\in V_{1},v\in V_{2}\}$.

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 co-graphs and threshold graphs. These trees/sequences can be used for algorithmic properties of those graphs.

4.1 Directed Co-Graphs

Definition 4.1** **(Directed Co-Graphs [BdGR97])

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

1. (i)

Every digraph on a single vertex $(\{v\},\emptyset)$, denoted by $\bullet$, is a directed co-graph. 2. (ii)

*If $G_{1}$ and $G_{2}$ are directed co-graphs, then *

(a) $G_{1}\oplus G_{2}$, * (b) $G_{1}\otimes G_{2}$, and * (c) $G_{1}\oslash G_{2}$*are directed co-graphs. * **

The class of directed co-graphs is denoted by DC.

The recursive definition of directed and undirected co-graphs lead to the following observation.

Observation 4.2

For every directed co-graph $G$ the underlying undirected graph ${\it un}(G)$ is a co-graph.

The reverse direction only holds under certain conditions, see Theorem 4.4.

Obviously for every directed co-graph we can define a tree structure, denoted as binary 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 binary di-co-tree in linear time, see [CP06].

In [BJM14] it is shown that the weak $k$-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 semi complete 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 [Gur17]. Calculs of directed co-graphs were also considered in connection with pomset logic in [Ret99]. Further the directed path-width, directed tree-width, directed feedback vertex set number, cycle rank, DAG-depth and DAG-width can be computed in linear time for directed co-graphs [GKR19b].

Lemma 4.3** **([GR18a])

Let $G$ be some digraph, then the following properties hold.

Digraph $G$ is a directed co-graph if and only if digraph $\text{co-}G$ is a directed co-graph. 2. 2.

Digraph $G$ is a directed co-graph if and only if digraph $G^{c}$ is a directed co-graph.

It further hold the following properties for directed co-graphs:

Theorem 4.4

Let $G$ be a digraph. The following properties are equivalent:

$G$* is a directed co-graph.* 2. 2.

$G\in\text{Free}(\{D_{1},\dots,D_{8}\})$. 3. 3.

$G\in\text{Free}(\{D_{1},\dots,D_{6}\})$* and ${\it un}(G)\in\text{Free}(\{P_{4}\})$.* 4. 4.

$G\in\text{Free}(\{D_{1},\dots,D_{6}\})$* and ${\it un}(G)$ is a co-graph.* 5. 5.

$G$* has directed NLC-width $1$.444For a definition of directed NLC-width, see [GWY16].* 6. 6.

$G$* has directed clique-width at most $2$ and $G\in\text{Free}(\{D_{2},D_{3}\})$.555For a definition of directed clique-width, see [CO00].*

Proof.

$(\ref{td-1})\Leftrightarrow(\ref{td-2})$ By [CP06].

$(\ref{td-2aa})\Leftrightarrow(\ref{td-2a})$ Since $\text{Forb}(\text{C})=\{P_{4}\}$.

$(\ref{td-1})\Leftrightarrow(\ref{td-3})$ By [GWY16]

$(\ref{td-1})\Leftrightarrow(\ref{td-4})$ By [GWY16]

$(\ref{td-2})\Rightarrow(\ref{td-2a})$ By Observation 4.2.

$(\ref{td-2aa})\Rightarrow(\ref{td-2})$ by Observation 2.6. ∎

For subclasses of directed co-graphs, 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:

$\{D_{1},\ldots,D_{8}\}=\text{co-}\{D_{1},\ldots,D_{8}\}$. 2. 2.

$\{D_{12},\ldots,D_{15}\}=\text{co-}\{D_{12},\ldots,D_{15}\}$.

For directed co-graphs Observation 4.5 leads to the next result.

Proposition 4.6

$\text{DC}=\text{co-}\text{DC}$.

4.2 Oriented Co-Graphs

Beside directed co-graphs and their subclasses we also will restrict the these classes to oriented graphs by omitting the series operation.

Definition 4.7** **(Oriented Co-Graphs)

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

1. (i)

Every digraph on a single vertex $(\{v\},\emptyset)$, denoted by $\bullet$, is an oriented co-graph. 2. (ii)

*If $G_{1}$ and $G_{2}$ are oriented co-graphs, then *

(a) $G_{1}\oplus G_{2}$*and * (b) $G_{1}\oslash G_{2}$*are oriented co-graphs. * **

The class of oriented co-graphs is denoted by OC.

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

Observation 4.8

For every oriented co-graph $G$ the underlying undirected graph ${\it un}(G)$ is a co-graph.

The reverse direction only holds under certain conditions, see Theorem 4.10. The class of oriented co-graphs was already analyzed by Lawler in [Law76] and [CLSB81, Section 5] using the notation of transitive series parallel (TSP) digraphs. A digraph $G=(V,A)$ is called transitive if for every pair $(u,v)\in A$ and $(v,w)\in A$ of arcs with $u\neq w$ the arc $(u,w)$ also belongs to $A$. For oriented co-graphs the oriented chromatic number and also the graph isomorphism problem can be solved in linear time [GKR19c, GKL20].

Lemma 4.9

Let $G$ be a digraph such that $G\in\text{Free}(\{\overleftrightarrow{K_{2}},D_{1},D_{5}\})$, then $G$ is transitive.

Proof.

Let $(u,v),(v,w)\in A$ be two arcs of $G=(V,A)$. Since $G\in\text{Free}(\{\overleftrightarrow{K_{2}}\})$ we know that $(v,u),(w,v)\not\in A$. Further since $G\in\text{Free}(\{D_{1},D_{5}\})$ we know that $u$ and $w$ are connected either only by $(u,w)\in A$ or by $(u,w)\in A$ and $(w,u)\in A$, which implies that $G$ is transitive. ∎

The class OC can also be defined by forbidden subdigraphs.

Theorem 4.10

Let $G$ be a digraph. The following properties are equivalent:

$G$* is an oriented co-graph* 2. 2.

$G\in\text{Free}(\{D_{1},D_{5},D_{8},\overleftrightarrow{K_{2}}\})$. 3. 3.

$G\in\text{Free}(\{D_{1},D_{5},\overleftrightarrow{K_{2}}\})$* and ${\it un}(G)\in\text{Free}(\{P_{4}\})$.* 4. 4.

$G\in\text{Free}(\{D_{1},D_{5},\overleftrightarrow{K_{2}}\})$* and ${\it un}(G)$ is a co-graph.* 5. 5.

$G$* directed has NLC-width $1$ and $G\in\text{Free}(\{\overleftrightarrow{K_{2}}\})$.* 6. 6.

$G$* has directed clique-width at most $2$ and $G\in\text{Free}(\{\overleftrightarrow{K_{2}}\})$.* 7. 7.

$G$* is transitive and $G\in\text{Free}(\{\overleftrightarrow{K_{2}},D_{8}\})$.*

Proof.

$(\ref{ch-oc1})\Rightarrow(\ref{ch-oc2})$ If $G$ is an oriented co-graph, then $G$ is a directed co-graph and by Theorem 4.4 it holds that $G\in\text{Free}(\{D_{1},\ldots,D_{8}\})$. Further $G\in\text{Free}(\{\overleftrightarrow{K_{2}}\})$ because of the missing series composition. This leads to $G\in\text{Free}(\{D_{1},D_{5},D_{8},\overleftrightarrow{K_{2}}\})$.

$(\ref{ch-oc2})\Rightarrow(\ref{ch-oc1})$ If $G\in\text{Free}(\{D_{1},D_{5},D_{8},\overleftrightarrow{K_{2}}\})$ then $G\in\text{Free}(\{D_{1},\ldots,D_{8}\})$ and is a directed co-graph. Since $G\in\text{Free}(\{\overleftrightarrow{K_{2}}\})$ there is no series operation in any construction of $G$ which implies that $G$ is an oriented co-graph.

$(\ref{ch-oc3a})\Leftrightarrow(\ref{ch-oc3})$ Since $\text{Forb}(\text{C})=\{P_{4}\}$.

$(\ref{ch-oc2})\Rightarrow(\ref{ch-oc6})$ By Lemma 4.9 we know that $G$ is transitive.

$(\ref{ch-oc6})\Rightarrow(\ref{ch-oc2})$ If $G$ is transitive, then $G\in\text{Free}(\{D_{1},D_{5}\})$.

$(\ref{ch-oc1})\Leftrightarrow(\ref{ch-oc4})$ and $(\ref{ch-oc1})\Leftrightarrow(\ref{ch-oc5})$ By Theorem 4.4.

$(\ref{ch-oc2})\Rightarrow(\ref{ch-oc3})$ By Observation 4.8.

$(\ref{ch-oc3a})\Rightarrow(\ref{ch-oc2})$ by Observation 2.6. ∎

Observation 4.11

Every oriented co-graph is a DAG.

Theorem 4.12** **([CLSB81])

A graph $G$ is a co-graph if and only if there exists an orientation $G^{\prime}$ of $G$ such that $G^{\prime}$ is an oriented co-graph.

4.3 Series-parallel partial order digraphs

We recall the definitions of from [BJG18] which are based on [VTL82]. A series-parallel partial order is a partially ordered set $(X,\leq)$ that is constructed by the series composition and the parallel composition operation starting with a single element.

• •

Let $(X_{1},\leq)$ and $(X_{2},\leq)$ be two disjoint series-parallel partial orders, then distinct elements $x,y\in X_{1}\cup X_{2}$ of a series composition666Note that the series composition in this case corresponds to the order composition in the definition of directed co-graphs. have the same order they have in $X_{1}$ or $X_{2}$. Respectively, this holds if both of them are from the same set, and $x\leq y$, if $x\in X_{1}$ and $y\in X_{2}$.

• •

Two elements $x,y\in X_{1}\cup X_{2}$ of a parallel composition are comparable if and only if both of them are in $X_{1}$ or both in $X_{2}$, while they keep their corresponding order.

Definition 4.13** **(Series-parallel partial order digraphs)

A series-parallel partial order digraph $G=(V,E)$ is a digraph, where $(V,\leq)$ is a series-parallel partial order and $(u,v)\in E$ if and only if $u\neq v$ and $u\leq v$.

The class of series-parallel partial order digraphs is denoted by SPO.

For a digraph $G=(V,E)$ an edge $(u,v)\in E$ is symmetric, if $(v,u)\in E$. 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 $G$ as $sym(G)$, which is the spanning subdigraph of $G$ that contains exactly the symmetric arcs of $G$. Analogously we define the asymmetric part of $G$ as $asym(G)$, which is the spanning subdigraph with only asymmetric edges.

Moreover, Bechet et al. showed in [BdGR97] the following property of directed co-graphs.

Lemma 4.14** **([BdGR97])

For every directed co-graph $G$ it holds that the asymmetric part of $G$ is a series-parallel partial order digraph and for the symmetric part the underlying undirected graph a co-graph.

The class of series-parallel partial ordered digraphs is equal to the class of oriented co-graphs, since they have exactly the same recursive structure. Thus this lemma is easy to prove with the following idea. Let $G$ be a directed co-graph and $T_{G}$ its corresponding di-co-tree.

• •

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

• •

asymmetric part: Exchange each series composition with a directed union composition. Since there are no more bidirectional edges left, this tree represents an oriented co-graph, e.g. a series-parallel 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.

1. (i)

Every digraph on a single vertex $(\{v\},\emptyset)$, denoted by $\bullet$, is a directed trivially perfect graph. 2. (ii)

If $G_{1}$ and $G_{2}$ are directed trivially perfect graphs, then $G_{1}\oplus G_{2}$ is a directed trivially perfect graph. 3. (iii)

*If $G$ is a directed trivially perfect graph, then *

(a) $G\oslash\bullet$, * (b) $\bullet\oslash G$, and * (c) $G\otimes\bullet$*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 $G$ the underlying undirected graph ${\it un}(G)$ is a trivially perfect graph.

The reverse direction only holds under certain conditions, see Theorem 4.18.

Lemma 4.17** **([GRR18])

For every digraph $G$ the following statements are equivalent.

$G$* is a transitive tournament.* 2. 2.

$G$* is an acyclic tournament.* 3. 3.

$G\in\text{Free}(\{\overrightarrow{C_{3}}\})$* and $G$ is a tournament.* 4. 4.

$G$* can be constructed from the one-vertex graph $K_{1}$ by repeatedly adding an out-dominating vertex.* 5. 5.

$G$* can be constructed from the one-vertex graph $K_{1}$ by repeatedly adding an in-dominated vertex.*

The class DTP can also be defined by forbidden induced subdigraphs. It holds that

Theorem 4.18

Let $G$ be a digraph. The following properties are equivalent:

$G$* is a directed trivially perfect graph.* 2. 2.

$G\in\text{Free}(\{D_{1},\dots,D_{15}\})$. 3. 3.

$G\in\text{Free}(\{D_{1},D_{2},D_{3},D_{4},D_{5},D_{6},D_{10},D_{11},D_{13},D_{14},D_{15}\})$* and ${\it un}(G)\in\text{Free}(\{C_{4},P_{4}\})$.* 4. 4.

$G\in\text{Free}(\{D_{1},D_{2},D_{3},D_{4},D_{5},D_{6},D_{10},D_{11},D_{13},D_{14},D_{15}\})$* and ${\it un}(G)$ is a trivially perfect graph.*

Proof.

$(\ref{ch-dtp1})\Rightarrow(\ref{ch-dtp2})$ The given forbidden digraphs $D_{1},\dots,D_{15}$ are not directed trivially perfect graphs and the set of directed trivially perfect graphs is closed under taking induced subdigraphs.

$(\ref{ch-dtp2})\Rightarrow(\ref{ch-dtp1})$ Since $G\in\text{Free}(\{D_{1},D_{2},D_{3},D_{4},D_{5},D_{6},D_{7},D_{8}\})$ digraph $G$ is a directed co-graph by [CP06] and thus has a construction using disjoint union, series composition, and order composition.

Since $G\in\text{Free}(\{D_{9},D_{10},D_{11}\})$ 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 $G\in\text{Free}(\{D_{12},D_{13},D_{14},D_{15}\})$ 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 $G\in\text{Free}(\{D_{5}\})=\text{Free}(\{C_{3}\})$ 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.

$(\ref{ch-dtp4})\Leftrightarrow(\ref{ch-dtp3})$ Since $\text{Forb}(\text{TP})=\{C_{4},P_{4}\}$.

$(\ref{ch-dtp2})\Rightarrow(\ref{ch-dtp3})$ By Observation 4.16.

$(\ref{td-2aa})\Rightarrow(\ref{td-2})$ By Observation 2.6. ∎

For directed trivially perfect graphs Observation 4.5 leads to the next result.

Proposition 4.19

$\text{DTP}\neq\text{co-}\text{DTP}$**

This motivates us to consider the class of edge complements of directed trivially perfect graphs.

Definition 4.20** **(Directed co-trivially perfect graphs)

The class of directed co-trivially perfect graphs is recursively defined as follows.

1. (i)

Every digraph on a single vertex $(\{v\},\emptyset)$, denoted by $\bullet$, is a directed co-trivially perfect graph. 2. (ii)

If $G_{1}$ and $G_{2}$ are directed co-trivially perfect graphs, then $G_{1}\otimes G_{2}$ is a directed trivially perfect graph. 3. (iii)

*If $G$ is a directed co-trivially perfect graph, then *

(a) $G\oslash\bullet$, * (b) $\bullet\oslash G$, and * (c) $G\oplus\bullet$*are directed co-trivially perfect graphs. * **

The class of directed co-trivially perfect graphs is denoted by DCTP.

Theorem 4.18 and Lemma 2.4 lead to the following characterization for directed co-trivially perfect graphs.

Theorem 4.21

Let $G$ be a digraph. The following properties are equivalent:

$G$* is a directed co-trivially perfect graph.* 2. 2.

$G\in\text{Free}(\{D_{1},\dots,D_{8},\text{co-}D_{11},\text{co-}D_{10},\text{co-}D_{9},D_{12},\ldots,D_{15}\})$. 3. 3.

$G\in\text{Free}(\{D_{1},\ldots,D_{6},D_{12},\ldots,D_{15}\})$* and ${\it un}(G)\in\text{Free}(\{P_{4},2K_{2}\})$.* 4. 4.

$G\in\text{Free}(\{D_{1},\ldots,D_{6},D_{12},\ldots,D_{15}\})$* and ${\it un}(G)$ is a co-trivially 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.

1. (i)

Every digraph on a single vertex $(\{v\},\emptyset)$, denoted by $\bullet$, is an oriented trivially perfect graph. 2. (ii)

If $G_{1}$ and $G_{2}$ are oriented trivially perfect graphs, then $G_{1}\oplus G_{2}$ is an oriented trivially perfect graph. 3. (iii)

*If $G$ is an oriented trivially perfect graph, then *

(a) $G\oslash\bullet$*and * (b) $\bullet\oslash G$*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 $G$ the underlying undirected graph ${\it un}(G)$ is a trivially perfect graph.

Similar as for oriented co-graphs we obtain a definition of OTP by forbidden induced subdigraphs.

Theorem 4.24

Let $G$ be a digraph. The following properties are equivalent:

$G$* is an oriented trivially perfect graph.* 2. 2.

$G\in\text{Free}(\{D_{1},D_{5},D_{8},D_{12},\overleftrightarrow{K_{2}}\})$. 3. 3.

$G\in\text{Free}(\{D_{1},D_{5},\overleftrightarrow{K_{2}}\})$* and ${\it un}(G)\in\text{Free}(\{C_{4},P_{4}\})$.* 4. 4.

$G\in\text{Free}(\{D_{1},D_{5},\overleftrightarrow{K_{2}}\})$* and ${\it un}(G)$ is a trivially perfect graph.* 5. 5.

$G$* is transitive and $G\in\text{Free}(\{\overleftrightarrow{K_{2}},D_{8},D_{12}\})$.*

Proof.

$(\ref{ch-otp1})\Rightarrow(\ref{ch-otp2})$ If $G$ is an oriented trivially perfect graph, then $G$ is a directed trivially perfect graph and $G\in\text{Free}(\{D_{1},\dots,D_{15}\})$. Further $G\in\text{Free}(\{\overleftrightarrow{K_{2}}\})$ because of the missing series composition. This leads to $G\in\text{Free}(\{D_{1},D_{5},D_{8},D_{12},\overleftrightarrow{K_{2}}\})$.

$(\ref{ch-otp2})\Rightarrow(\ref{ch-otp1})$ If $G\in\text{Free}(\{D_{1},D_{5},D_{8},D_{12},\overleftrightarrow{K_{2}}\})$, then $G\in\text{Free}(\{D_{1},\dots,D_{15}\})$ and is a directed trivially perfect graph. Since $G\in\text{Free}(\{\overleftrightarrow{K_{2}}\})$ there is no series operation in any construction of $G$ which implies that $G$ is an oriented trivially perfect graph.

$(\ref{ch-otp3a})\Leftrightarrow(\ref{ch-otp3})$ Since $\text{Forb}(\text{TP})=\{C_{4},P_{4}\}$.

$(\ref{ch-otp2})\Rightarrow(\ref{ch-otp3})$ By Observation 4.23.

$(\ref{ch-otp3a})\Rightarrow(\ref{ch-otp2})$ By Observation 2.6

$(\ref{ch-otp2})\Rightarrow(\ref{ch-otp4})$ By Lemma 4.9 we know that $G$ is transitive.

$(\ref{ch-otp4})\Rightarrow(\ref{ch-otp2})$ If $G$ is transitive, then $G$ has no induced $D_{1},D_{5}$. ∎

Theorem 4.25

A graph $G$ is a trivially perfect graph if and only if there exists an orientation $G^{\prime}$ of $G$ such that $G^{\prime}$ is an oriented trivially perfect graph.

Proof.

Let $G$ be a trivially perfect graph. Then $G$ is also a comparability graph, which implies that $G$ has a transitive orientation $G^{\prime}$. Since $G\in\text{Free}(\{C_{4},P_{4}\})$ it follows that $G^{\prime}\in\text{Free}(\{D_{8},D_{12}\})$. Further by definition $G^{\prime}\in\text{Free}(\{\overleftrightarrow{K_{2}}\})$. By Theorem 4.24 we know that $G^{\prime}$ is an oriented trivially perfect graph.

For the reverse direction let $G^{\prime}$ be an oriented trivially perfect graph. Then by Theorem 4.24 it holds that $G^{\prime}\in\text{Free}(\{D_{8},D_{12}\})$. Since $D_{8}$ is the only transitive orientation of $P_{4}$ and $D_{12}$ is the only transitive orientation of $C_{4}$ it holds that ${\it un}(G)\in\text{Free}(\{C_{4},P_{4}\})$. Thus ${\it un}(G)$ is a trivially perfect graph. ∎

Observation 4.26

If $G\in DTP$ then the underlying undirected graph of the symmetric part of $G$ is trivially perfect and the asymmetric part of $G$ 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 co-trivially perfect graphs)

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

1. (i)

Every digraph on a single vertex $(\{v\},\emptyset)$, denoted by $\bullet$, is an oriented co-trivially perfect graph. 2. (ii)

*If $G$ is an oriented co-trivially perfect graph, then *

(a) $G\oslash\bullet$, * (b) $\bullet\oslash G$, and * (c) $G\oplus\bullet$*are oriented co-trivially perfect graphs. * **

The class of oriented co-trivially perfect graphs is denoted by OCTP.

Restricting the operations of directed co-trivially 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.

1. (i)

Every edgeless digraph is a directed weakly quasi threshold graph. 2. (ii)

If $G_{1}$ and $G_{2}$ are directed weakly quasi threshold graphs, then $G_{1}\oplus G_{2}$ is a directed weakly quasi threshold graph. 3. (iii)

*If $G$ is a directed weakly quasi threshold graph and $I$ is an edgeless digraph, then *

(a) $G\oslash I$, * (b) $I\oslash G$, and * (c) $G\otimes I$*are directed weakly quasi threshold graphs. * **

The class of directed weakly quasi threshold graphs is denoted by DWQT.

Observation 4.29

If $G$ is a directed weakly quasi threshold graph, ${\it un}(G)$ is weakly quasi threshold graph.