Provide evidence that the modified version of the choice problem has an NP-complete solution; Exists, given a graph G and a target cost c, a spanning tree in which the highest possible payment at any vertex does not exceed the target cost?

Given a vertex set W = {1, 2, 3, 4}, solve for the following questions: (i) Knowing that edges are the same if and only if they have the same endpoint, how many different edges are possible? (ii) Suppose for this question that graphs are different if they have different sets of edges (they do not depend on if the graphs are isomorphic or not). How many simple graphs are there with the vertex set W? (iii) Draw the eleven nonisomorphic simple graphs that have four vertices.

I need a proof as a solution for both of these questions.

Problem 2. Use Bellman-Ford to determine the shortest paths from vertex s to all other vertices in this graph. Afterwards, indicate to which vertices s has a well defined shortest path, and which do not by indicating the distance as -o∞. Draw the resulting shortest path tree containing the vertices with well defined shortest paths. For consistency, you should relax the edges in the following order: s → a, s → b, a → c, b → a, b → d, c → b, с — а, с —е, d — f, e — d and f — e. — а, b, а — с, b — а, b — d, с — b, → d and f -→e. a -1 3 5 -2 -5 S 4 4 10 6 b d f -10 -6