We consider two types of matroids defined on the edge set of a graph G: count matroids Mk,ℓ(G), in which independence is defined by a sparsity count involving the parameters k and ℓ, and the C21-cofactor matroid C(G), in which independence is defined by linear independence in the cofactor matrix of G. We show, for each pair (k,ℓ), that if G is sufficiently highly connected, then G−e has maximum rank for all e∈E(G), and the matroid Mk,ℓ(G) is connected. These results unify and extend several previous results, including theorems of Nash-Williams and Tutte (k=ℓ=1), and Lovász and Yemini (k=2,ℓ=3). We also prove that if G is highly connected, then the vertical connectivity of C(G) is also high.We use these results to generalize Whitney's celebrated result on the graphic matroid of G (which corresponds to M1,1(G)) to all count matroids and to the C21-cofactor matroid: if G is highly connected, depending on k and ℓ, then the count matroid Mk,ℓ(G) uniquely determines G; and similarly, if G is 14-connected, then its C21-cofactor matroid C(G) uniquely determines G. We also derive similar results for the t-fold union of the C21-cofactor matroid, and use them to prove that every 24-connected graph has a spanning tree T for which G−E(T) is 3-connected, which verifies a case of a conjecture of Kriesell.