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). 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.
Highlights
During the last years classes of directed graphs have received a lot of attention (BangJensen and Gutin 2018) since they are useful in multiple applications of directed networks
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
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
Summary
During the last years classes of directed graphs have received a lot of attention (BangJensen and Gutin 2018) since they are useful in multiple applications of directed networks. In order to get an overview of the classes, we compare the classes to each other and visualize the relations between them All of these classes are hereditary, just like directed cographs, such that they can be characterized by a set of forbidden induced subdigraphs. Undirected co-graphs, i.e. complement reducible graphs, appeared independently by several authors (see Lerchs 1971; Sumner 1974) for example, while directed cographs were introduced 30 years later by Bechet et al (1997) Due to their recursive structure there are problems, that are hard in general, which can be solved efficiently on (directed) co-graphs (see Bodlaender and Möhring 1993; Corneil et al 1981, 1984; Lin et al 1995; Gurski et al 2020b; Bang-Jensen and Maddaloni 2014; Gurski 2017; Gurski et al 2019b, c; Gurski and Rehs 2018b; Gurski et al 2020a, c).
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