When does a matroid $M$ have a set of circuits $\mathcal{C}$ with a connected intersection graph i.e. when is the graph $G$ with$V(G)=\mathcal{C}$ and adjacencies $\{A,B\}\in E(G)\iff A\cap B\neq\emptyset$ connected?


This is equivalent to charactering the matroids with a partial ear-decomposition i.e. the matroids with circuits that can be indexed $C\_1,\ldots C\_n$ so we get that $\forall 0<i\leq n\exists j<i:C\_i\cap C\_j\neq\emptyset$ (where note this indexing is not necessarily injective i.e. there might exist $i\neq j$ with $C\_i=C\_j$)




---


Suppose we call matroids with this property special now if two matroids $M\_1$ and $M\_2$ are special and some circuit in $M\_1$ is not disjoint to some circuit in $M\_2$ then $M\_1\oplus M\_2$ is also special, with that said then what do "special" matroids look like? Is there a simple way to characterise these?



When do the circuits of a matroid have a connected intersection graph?
This holds if and only if $M$ has at most one connected component which contains a circuit. Clearly, the intersection graph of circuits is disconnected if $M$ has two connected components which each contain a circuit. For the other direction, suppose that $M$ has at most one connected component $N$ which contains a circuit. If $M$ has at most one circuit, then clearly the intersection graph of circuits is connected. Otherwise, let $C\_1$ and $C\_2$ be distinct circuits of $M$. Note that $C\_1$ and $C\_2$ are circuits of $N$. Choose $e \in C\_1$ and $f \in C\_2$. Since $N$ is connected, there is a circuit $C\_3$ of $N$ such that $\{e,f\} \subseteq C\_3$. Thus, there is path of length $2$ between $C\_1$ and $C\_2$ in the intersection graph of circuits.


