  We say that a graph $G$ is $(2,m)$-linked if, for any distinct vertices
$a_1,\ldots, a_m, b_1,b_2$ in $G$, there exist vertex disjoint connected
subgraphs $A,B$ of $G$ such that $\{a_1, \ldots, a_m\}$ is contained in $A$ and
$\{b_1,b_2\}$ is contained in $B$. A fundamental result in structural graph
theory is the characterization of $(2,2)$-linked graphs, with different
versions obtained independently by Robertson and Chakravarty, Seymour, and
Thomassen. It appears to be very difficult to characterize $(2,m)$-linked
graphs for $m\ge 3$. In this paper, we provide a partial characterization of
$(2,m)$-linked graphs by adding an average degree condition. This implies that
$(2m+2)$-connected graphs are $(2,m)$-linked. Moreover, if $G$ is a
$(2m+2)$-connected graph and $a_1, \ldots, a_m, b_1,b_2$ are distinct vertices
of $G$, then there is a path $P$ in $G$ between $b_1$ and $b_2$ and avoiding
$\{a_1, \ldots, a_m\}$ such that $G-P$ is connected, improving a previous
connectivity bound of $10m$.
