Hierarchical graph

Hierarchical graph --- иерархический граф.

Let [math]\displaystyle{ G }[/math] be a graph of some type, e.g. G can be an undirected graph, a digraph or a hypergraph, and let [math]\displaystyle{ F }[/math] be a set of its fragments such that [math]\displaystyle{ G\in F }[/math] and, for any [math]\displaystyle{ C_1, C_2 \in F }[/math], just one of the following properties holds: (1) [math]\displaystyle{ C_1\subset C_2 }[/math], (2) [math]\displaystyle{ C_2\subset C_1 }[/math], (3) [math]\displaystyle{ C_1 \cap C_2= \varnothing }[/math].

[math]\displaystyle{ H = (G,T) }[/math], where [math]\displaystyle{ T=(F,I) }[/math] is a directed tree with a root [math]\displaystyle{ G }[/math] such that [math]\displaystyle{ I }[/math] represents an immediate inclusion relation between fragments of [math]\displaystyle{ F }[/math], is called a hierarchical graph; [math]\displaystyle{ G }[/math] is called the underlying graph of [math]\displaystyle{ H }[/math], and [math]\displaystyle{ T }[/math] is called the inclusion tree of [math]\displaystyle{ H }[/math].

A hierarchical graph H is called a connected one, if each fragment from [math]\displaystyle{ F }[/math] is a connected graph, and a simple one, if all fragments from [math]\displaystyle{ F }[/math] are induced subgraphs of [math]\displaystyle{ G }[/math].

A simple hierarchical graph [math]\displaystyle{ H=(G, T) }[/math] such that [math]\displaystyle{ G }[/math] is an undirected graph and the leaves of [math]\displaystyle{ T }[/math] are exactly the trivial subgraphs of [math]\displaystyle{ G }[/math] is called a clustered graph.