Comparability graph

Comparability graph --- граф сравнимости.

Let [math]\displaystyle{ G = (V,E) }[/math] be an undirected graph and let [math]\displaystyle{ F }[/math] be an orientation of its edges (i.e. [math]\displaystyle{ (V,F) }[/math] is the resulting oriented graph). [math]\displaystyle{ F }[/math] is called a transitive orientation of [math]\displaystyle{ G }[/math] if the following properties hold:

[math]\displaystyle{ F \cap F^{-1} = \emptyset\mbox{ and }F + F^{-1} = E\mbox{ and }F^{2} \subseteq F, }[/math]

where [math]\displaystyle{ F^{2} = \{(a,c)| \; \exists_{b \in V}(a,b) \in F \wedge (b,c) \in F\} }[/math].

A graph [math]\displaystyle{ G }[/math] which admits a transitive orientation of its edges is called a comparability graph.

If graph [math]\displaystyle{ G }[/math] is a comparability graph, then this also holds for every induced subgraph of [math]\displaystyle{ G }[/math].

The other name is a Transitively orientable graph.