We consider two types of matroids defined on the edge set of a graph G: count matroids ${\mathcal{M}}_{k,\ell }(G)$, in which independence is defined by a sparsity count involving the parameters k and ℓ, and the $C_2^1$-cofactor matroid $\mathcal{C}(G)$, 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 $e\in E(G)$, and the matroid ${\mathcal{M}}_{k,\ell }(G)$ 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 $\mathcal{C}(G)$ is also high.

We use these results to generalize Whitney's celebrated result on the graphic matroid of G (which corresponds to ${\mathcal{M}}_{1,1}(G)$) 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 ${\mathcal{M}}_{k,\ell }(G)$ uniquely determines G; and similarly, if G is 14-connected, then its $C_2^1$-cofactor matroid $\mathcal{C}(G)$ 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 $G-E(T)$ is 3-connected, which verifies a case of a conjecture of Kriesell.

我们考虑在图G的边集上定义的两种类型的拟阵：计数拟阵${\mathcal{中号}}_{k,\ell }（G）$，其中独立性由涉及参数k和ℓ的稀疏性计数定义，并且$C_2^1$-辅助因子拟阵$\mathcal{C}（G）$，其中独立性由G的辅因子矩阵中的线性独立性定义。我们展示，对于每一对$（k,\ell ）$，如果G足够高度连通，则$G-e$所有人的排名最高$e\epsilon 乙（G）$，和拟阵${\mathcal{中号}}_{k,\ell }（G）$已连接。这些结果统一并扩展了之前的几个结果，包括 Nash-Williams 和 Tutte 定理（$k=\ell =1$）、Lovász 和 Yemini（$k=2,\ell =3$）。我们还证明，如果G是高度连通的，则$\mathcal{C}（G）$也很高。

我们使用这些结果来概括 Whitney 在G的图形拟阵上的著名结果（对应于${\mathcal{中号}}_{1,1}（G）$）到所有计数拟阵和$C_2^1$-cofactor matroid：如果G是高度连通的，取决于k和ℓ，则计数矩阵${\mathcal{中号}}_{k,\ell }（G）$唯一确定G；类似地，如果G是 14 连通的，那么它的$C_2^1$-辅助因子拟阵$\mathcal{C}（G）$唯一确定G。我们还对t折叠并集得出了类似的结果$C_2^1$-cofactor matroid，并用它们证明每个24连通图都有一个生成树T，其中$G-乙（\mathrm{时间}）$是3连通的，验证了Kriesell猜想的一个例子。