Vertex path cover

Vertex path cover --- вершинное путевое покрытие.

Let [math]\displaystyle{ {\mathcal P} = \{P_{1}, \ldots, P_{k}\} }[/math] be a set of paths in a digraph [math]\displaystyle{ D }[/math]. [math]\displaystyle{ {\mathcal P} }[/math] is a vertex path cover of [math]\displaystyle{ D }[/math] iff [math]\displaystyle{ \{V(P_{1}), \ldots, V(P_{k})\} }[/math] is a partition of [math]\displaystyle{ V(D) }[/math].

[math]\displaystyle{ pn_{v}(D) = \min\{|{\mathcal P}|: {\mathcal P}\mbox{ is a vertex path cover of }D\} }[/math]

is the vertex path number of [math]\displaystyle{ D }[/math].