Consider a connected undirected graph G = (V, E) with distinct positive edge weights wG(e) > 0. Consider another graph H with the same vertex and edge sets, but wH (e) = wG(e) + 1 for every edge e E.

(a) Do G and H have the same minimum spanning tree? Prove that they do or provide a counter-example demonstrating that they do not.

(b) Do G and H have the same shortest paths between all pairs of vertices u, v V ? Prove that they do or provide a counter-example demonstrating that they do not.