For a matroid $N$, a matroid $M$ is $N$-connected if every two elements of $M$ are in an $N$-minor together. Thus a matroid is connected if and only if it is $U_{ 1,2}$-connected. This paper proves that $U_{ 1,2}$ is the only connected matroid $N$ such that if $M$ is $N$-connected with $|E(M)| > |E(N)|$, then $M \backslash e$ or $M / e$ is $N$-connected for all elements $e$. Moreover, we show that $U_{ 1,2}$ and $M(\mathcal{ W}_2)$ are the only connected matroids $N$ such that, whenever a matroid has an $N$-minor using $\{ e,f\}$ and an $N$-minor using $\{ f,g\}$, it also has an $N$-minor using $\{ e,g\}$. Finally, we show that $M$ is $U_{ 0,1} \oplus U_{ 1,1}$-connected if and only if every clonal class of $M$ is trivial.
Categories: math.CO