Edge path cover

Edge path cover --- дуговое 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 an edge path cover of [math]\displaystyle{ D }[/math] iff [math]\displaystyle{ \{E(P_{1}), \ldots, E(P_{k})\} }[/math] is a partition of [math]\displaystyle{ E(D) }[/math].

[math]\displaystyle{ pn_{e}(D) = \min\{|{\mathcal P}|: {\mathcal P}\mbox{ is an edge path cover of }D\} }[/math]

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