  Let H = (H,V) be a hypergraph with edge set H and vertex set V. Then
hypergraph H is invertible iff there exists a permutation pi of V such that for
all E belongs to H(edges) intersection of(pi(E) and E)=0. H is invertibility
critical if H is not invertible but every hypergraph obtained by removing an
edge from H is invertible. The degree of H is d if |{E belongs to H(edges)|x
belongs to E}| =< d for each x belongs to V Let i(d) be the maximum number of
edges of an invertibility critical hypergraph of degree d. Theorem: i(d) =<
(d-1) {2d-1 choose d} + 1. The proof of this result leads to the following
covering problem on graphs: Let G be a graph. A family H is subset of (2^{V(G)}
is an edge cover of G iff for every edge e of G, there is an E belongs to
H(edge set) which includes e. H(edge set) is a minimal edge cover of G iff for
H' subset of H, H' is not an edge cover of G. Let b(d) (c(d)) be the maximum
cardinality of a minimal edge cover H(edge set) of a complete bipartite graph
(complete graph) where H(edge set) has degree d. Theorem: c(d)=< i(d)=<b(d)=<
c(d+1) and 3. 2^{d-1} - 2 =< b(d)=< (d-1) {2d-1choose d} +1. The proof of this
result uses Sperner theory. The bounds b(d) also arise as bounds on the maximum
number of elements in the union of minimal covers of families of sets.
