Let G be a simple connected graph. Let a, b, c, d be four distinct vertices of G.
Is there a way to partition the above four vertices to two pairs, so that the two shortest paths between the vertices of each pair are edgedisjoint?
Let G be a simple connected graph. Let a, b, c, d be four distinct vertices of G. Is there a way to partition the above four vertices to two pairs, so that the two shortest paths between the vertices of each pair are edgedisjoint? 


Yes. Just pick the two paths (not necessarily edge disjoint) in G of shortest total length which together join the four vertices into two pairs. If they contained a common edge, you could remove that edge from both paths (changing which vertices are connected to which) to obtain a pair of paths of shorter total length. The resulting paths must be shortest paths between the pairs of vertices they connect. 

