Consider the following property of an undirected graph: For any four distinct vertices $a,b,c,d$, there is a path from $a$ to $b$ and a path from $c$ to $d$ such that the two paths do not share any vertex.


Is there a name for this property, or has it been studied? It looks related to vertex connectivity but not quite the same. Also it seems that any graph satisfying this property must have vertex connectivity at least $2$.



Disjoint paths between four vertices
The property that you are describing is called *$2$-linked*. More generally, we say that a graph is *$k$-linked* if it has at least $2k$ vertices and for all distinct vertices $s\_1, \dots s\_k, t\_1, \dots, t\_k$ there are $k$-vertex disjoint paths connecting $s\_i$ to $t\_i$ for all $i \in [k]$. Note that every $k$-linked graph is $k$-connected. The converse is not true. For example, a cycle is $2$-connected but is not $2$-linked.


However, it is a classic theorem that there is a function $f(k)$ such that every $f(k)$-connected graph is $k$-linked. The current best bound for $f(k)$ is due to Thomas and Wollan, where they prove that every $10k$-connected graph is $k$-linked. The paper is titled *An improved linear edge bound for graph linkages*, and is available [here](https://core.ac.uk/download/pdf/82634043.pdf).


