F-Width (of a hypergraph)

[math]\displaystyle{ F }[/math]-Width (of a hypergraph) --- [math]\displaystyle{ F }[/math]-ширина.

Let [math]\displaystyle{ H }[/math] and [math]\displaystyle{ F }[/math] be two hypergraphs on the same vertex set. The [math]\displaystyle{ F }[/math]-width [math]\displaystyle{ w(H,F) }[/math] of [math]\displaystyle{ H }[/math] is the minimal size of an [math]\displaystyle{ H }[/math]-covering set of edges from [math]\displaystyle{ F }[/math]. The [math]\displaystyle{ F }[/math]-matching width [math]\displaystyle{ mw(H,F) }[/math] of [math]\displaystyle{ H }[/math] is the maximum, over all matchings [math]\displaystyle{ M }[/math] in [math]\displaystyle{ H }[/math], of [math]\displaystyle{ w(H,F) }[/math]. We write [math]\displaystyle{ w(H) }[/math] and [math]\displaystyle{ mw(H) }[/math] for [math]\displaystyle{ w(H,H) }[/math] and [math]\displaystyle{ mw(H,H) }[/math], respectively, and call these parameters the width and matching width of [math]\displaystyle{ H }[/math].

The independent [math]\displaystyle{ F }[/math]-width [math]\displaystyle{ iw(H, F) }[/math] of [math]\displaystyle{ H }[/math] is the minimal size of a covering matching in [math]\displaystyle{ F }[/math]. The independent [math]\displaystyle{ F }[/math]-matching width [math]\displaystyle{ imv(H,F) }[/math] of [math]\displaystyle{ H }[/math] is the maximum, over all matchings [math]\displaystyle{ M }[/math] of [math]\displaystyle{ H }[/math], of [math]\displaystyle{ im(M,F) }[/math]. Again, we write [math]\displaystyle{ iw(H) }[/math] for [math]\displaystyle{ iw(H,H) }[/math] and [math]\displaystyle{ imw(H,H) }[/math], and call them the independent width and independent matching width of [math]\displaystyle{ H }[/math], respectively.