Define the optimization problem LONGEST-PATH-LENGTH as the relation that associates each instance of an undirected graph and two vertices with the number of edges in a longest simple path between the two vertices. Define the decision problem LONGEST-PATH = {〈G, u, ν, k〉 : G = (V, E) is an undirected graph, u, ν ∈ V, k ≥ 0 is an integer, and there exists a simple path from u to ν in G consisting of at least k edges}. Show that the optimization problem LONGEST-PATH-LENGTH can be solved in polynomial time if and only if LONGEST-PATH ∈ P.