Base decomposition of matroids I want to find a generalization of the idea that, in a graphic matroid, every base can be decomposed on the stars (edges adjacent to a vertex).
For example one could say that a matroid $M$ of rank $k$ has "generating independent sets" $I_1, \dots, I_k$ if every base $B$ of $M$ can be written $B = \{b_1, ..., b_k\}$ with $b_i \in I_i$.
I want to know if these matroids have been studied and if they have a name.
I found the notion of "combinatorial decomposable" (http://www.math.univ-montp2.fr/~ramirez/Oxford.pdf) which is close but not exactly the same.
There is also the Matroid covering theorem (http://math.mit.edu/~goemans/18438F09/lec13.pdf) but again it is not exactly the same.
I would also be interested in the stronger property that there is a unique way to write the base $B$ as a system of distinct representative in $I_1, \cdots , I_k$.
 A: Let $G = (V,E)$ be a connected graph on $n$ vertices. The star at vertex $v$ described in the question is the cut-set associated the partition $v \uplus V\setminus v$ of the vertices. We remark that these cut-sets will be independent sets when $G$ is a simple graph, but will not always be independent sets for general (multi)graphs. Take any $n-1$ vertices, the cut-sets associated the these vertices given a basis of the cut-space of $G$. The cut-space is the orthgonal complement of the cycle-space of $G$ in the edge-space which is $\mathbb{F}_2^E$ where addition is given by symmetric difference. We now give a generalization of the OP's observation that every spanning tree can be decomposed over the cut-sets of vertices.

If $T = \{e_1, \dots, e_{n-1}\}$ is a spanning tree and $C^*_1, \dots, C^*_{n-1}$ be a basis of the cut-space, then there exists $\sigma \in S_{n-1}$ such that $e_i \in C^*_{\sigma(i)}$.

The spanning tree $T$ gives us another basis $C^*_{e_1}, \dots, C^*_{e_{n-1}}$ of the cut-space consisting of fundamental cuts. That is, $C^*_{e_i}$ is the set of edges of $G$ crossing the partition of vertices given by the connected components of $T \setminus e$. We observe that $e_i \in C^*_{e_i}$ for all $i$ and $e_i \not\in C^*_{e_j}$ for all $i \ne j$.
Since the fundamental cuts form a basis we have
$$
\begin{bmatrix}
  & \vdots &\\
\cdots & a_{ij} & \cdots\\
  & \vdots & 
\end{bmatrix}
\begin{bmatrix}
C^*_{e_1}\\
\vdots \\
C^*_{e_{n-1}}
\end{bmatrix}
=
\begin{bmatrix}
C^*_1\\
\vdots \\
C^*_{n-1}
\end{bmatrix}
$$
for $A = [a_{ij}] \in \mathbb{F}_2^{(n-1) \times (n-1)}$. Notice $a_{ij} = 1$ if and only if $e_j \in C^*_i$. So, we must find a permutation $\sigma \in S_{n-1}$ such that $a_{i,\sigma(i)} = 1$ for all $i$.  The matrix $A$ is invertible since $C^*_1, \dots, C^*_{n-1}$ is also a basis. So,
$$\det A = \sum_{\sigma \in S_{n-1}} \prod_i a_{i,\sigma(i)} \ne 0$$
and it follows such a $\sigma$ exists.
Now this generalizes the OP's observation within the graphic matroid setting. For general matroids cut-sets become cocircuits. However, the cocircutis of general matroids do not form a vector space with symmetric difference. So, the argument above does not extend, but perhaps something can be said looking that cocircuits or fundamental cocircuits
For an example consider the non-graphic matroid $U_{2,4}$. The symmetric difference of the cociruits $123$ and $234$ is $14$ which is not a cocircuit but a base. However, the fundamental cociruits of $12$ are $134$ and $234$, and every base of $U_{2,4}$ can be decomposed over these cocircuits.
