In graph theory, there are two related properties of a hypergraph that are called its "width". Given a hypergraph "H" = ("V", "E"), we say that a set "K" of edges "pins" another set "F" of edges if every edge in "F" intersects some edge in "K". Then: 
Since "E" contains all matchings in "E", for all "H": w("H") ≥ mw("H").
The width of a hypergraph is used in Hall-type theorems for hypergraphs.
Examples.
Let "H" be the hypergraph with vertex set V = {A,B; a,b} and edge set:"E" = { {A,a}, {B,b}, {A,b}, {B,a} }The widths of "H" are: 
Characterizations.
The disjointness graph of "H", denoted D("H"), is a graph where each edge in H is a vertex in D("H"), and every two disjoint edges in "H" are adjacent in D("H"). The matchings in "H" correspond to the cliques in D("H"). Meshulam characterized the widths of a hypergraph "H" in terms of the properties of D("H"). For any positive integer "r":
The line graph of "H", denoted L("H"), is a graph where each edge in H is a vertex in L("H"), and every two intersecting edges in "H" are adjacent in L("H"). The matchings in H correspond to the independent sets in L("H"). Since L("H") is the complement of D("H"), the above characterization can be translated to L("H"):
The domination number of a graph "G", denoted "γ"("G"), is the smallest size of a vertex set that dominates all vertices of "G". The width of a hypergraph equals the domination number or its line-graph: w("H") = "γ"(L("H")). This is because the edges of "E" are the vertices of L("H"): every subset of "E" that pins "E" in "H" corresponds to a vertex set in L("H") that dominates all L("H").
The independence domination number of a graph "G", denoted "iγ"("G"), is the maximum, over all independent sets "A" of "G", of the smallest set dominating "A". The matching width of a hypergraph equals the independence domination number or its line-graph: mw("H") = "iγ"(L("H")). This is because every matching "M" in "H" corresponds to an independent set "IM" in L("H"), and every subset of "E" that pins "M" in "H" corresponds to a set that dominates "IM" in L("H").
References.
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