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 $2$.