If U and W are disjoint sets of vertices in a nonseparable g

If U and W are disjoint sets of vertices in a nonseparable graph G of order 4 or more with |U| = |W| = 2, then G contains two disjoint paths connecting the vertices of U and the vertices of W.

Solution

Let U={u1,u2} and W={w1,w2}. I know that G is non-separable, so every pair of vertices in G must have a common cycle. Every vertex in U have common cycle with every vertex in W, so there is a path uw for every pair of vertex u,w.

or

Let H be a graph obtain by adding 2 vertices a,b to G, then joining a to U and bb to W. Since G is non-separable, H is non-separable, thus every 2 vertices of H have common cycle. Therefore, I can find a path uw that goes through a,b.

On the first attempt, I found the path uw doesn\'t contain a,b now I found another path uwthat contain a,b. So these 2 paths are disjoint.