Let G = (V, E) be a directed graph. We say that a set of paths are non-overlapping if they have no edges in common. For example, the set of 3 (purple, orange, and green) paths in the graph below form a set of 3 non-overlapping paths. Note that it's okay for the vertices to overlap. (u.a, v ) ( u, biv ) ( U. C. b, d, v ) However, in the set of 3 paths drawn below, since the purple and green paths overlap along the edge (b, v), they do not form a non-overlapping set of paths. If we just take the orange path with either the purple or the green, however, we have a set of two non-overlapping paths. ( U, cib, v ) (u , a, b, v) ( U, e. u, b, a, ) Let u, v E V be two distinct vertices of the directed graph G. We say that a set of edges C C E disconnects v from u if and only if when we remove C from G we no longer have any path from u to v. (This is related to our notion of a bridge in an undirected graph from our lecture on the strength of weak ties.) For example, the 3 different sets (orange, purple, and green) of edges in the graph below each form a set that disconnects v from u, as does the union of any of these. { ( u , a ) , (u,b ), ( cob) } 8 ( aN) , ( bi ) (a,v)} fla, v ) , ( biv ), (bad)} Let P = {P1, P2, ..., Px} be any set of non-overlapping u-v paths, and let C be any set of edges whose removal disconnects v from u. Use a proof by contradiction and the Pigeonhole Prin- ciple to prove that [P|