Linkages and removable paths avoiding vertices

A graph G is $(2,m)$-linked if, for any distinct vertices $a_1,\dots ,a_m,b_1,b_2$ in G, there exist disjoint connected subgraphs $A,B$ of G such that $a_1,\dots ,a_m\in V\left(A\right)$ and $b_1,b_2\in V\left(B\right)$. A fundamental result in structural graph theory is the characterization of $(2,2)$-linked graphs. It appears to be difficult to characterize $(2,m)$-linked graphs for $m\ge 3$. In this paper, we provide a partial characterization of $(2,m)$-linked graphs. This implies that every $(2m+2)$-connected graphs G is $(2,m)$-linked and for any distinct vertices $a_1,\dots ,a_m,b_1,b_2$ of G, there is a path P in G between $b_1$ and $b_2$ and avoiding $\{a_1,\dots ,a_m\}$ such that $G-P$ is connected, improving a previous connectivity bound of 10m.