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). The class of 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 digraphs which can be defined by, starting with the single vertex graph, 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. The restriction to linear expressions was recently studied by Boeckner. Until now, there are only a few versions of subclasses of directed co-graphs. 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 give characterizations by finite sets of minimal forbidden induced subdigraphs. Additionally, we analyze the relations between these graph classes.

无向共形图是可以通过不相交联合和联接操作从单个顶点图生成的图。共同图完全是无\（P_4 \）图（其中\（P_4 \）表示在4个顶点上的路径）。合作图形本身及其几个子类已经得到了深入的研究。其中有一些平凡的理想图，阈值图，弱拟阈值图和简单的协同图。有向共形图是有向图，可以通过从单个顶点图开始，应用不相交的并集，有序组成和级数组成来定义有向图。通过省略系列组成，我们获得了定向共形图的子类，该类已由Lawler在1970年代进行了分析。Boeckner最近研究了对线性表达式的限制。到现在为止，有向共同图的子类只有几个版本。通过将无向子类的限制传递给有向类，我们为有向共同图定义了相应的子类。我们考虑阈值图，简单共同图，共同简单图，平凡完美图，平凡完美图，弱准阈值图和准弱准阈值图的有向和定向版本。对于所有这些类，我们通过最小的禁止诱导子图的有限集合给出特征。此外，我们分析了这些图类之间的关系。