Count and cofactor matroids of highly connected graphs

We consider two types of matroids defined on the edge set of a graph G: count matroids $M_{k,\ell }\left(G\right)$, in which independence is defined by a sparsity count involving the parameters k and ℓ, and the $C_2^1$-cofactor matroid $C\left(G\right)$, in which independence is defined by linear independence in the cofactor matrix of G. We show, for each pair $(k,\ell )$, that if G is sufficiently highly connected, then $G-e$ has maximum rank for all $e\in E\left(G\right)$, and the matroid $M_{k,\ell }\left(G\right)$ is connected. These results unify and extend several previous results, including theorems of Nash-Williams and Tutte ($k=\ell =1$), and Lovász and Yemini ($k=2,\ell =3$). We also prove that if G is highly connected, then the vertical connectivity of $C\left(G\right)$ is also high.

We use these results to generalize Whitney's celebrated result on the graphic matroid of G (which corresponds to $M_{1,1}\left(G\right)$) to all count matroids and to the $C_2^1$-cofactor matroid: if G is highly connected, depending on k and ℓ, then the count matroid $M_{k,\ell }\left(G\right)$ uniquely determines G; and similarly, if G is 14-connected, then its $C_2^1$-cofactor matroid $C\left(G\right)$ uniquely determines G. We also derive similar results for the t-fold union of the $C_2^1$-cofactor matroid, and use them to prove that every 24-connected graph has a spanning tree T for which $G-E\left(T\right)$ is 3-connected, which verifies a case of a conjecture of Kriesell.