Determine {G | G is a complete graph} is in P or NP-complete

Prove the choice problem variant is NP-complete; Exists a spanning tree with a goal cost c for a graph G and a vertex's maximum payment?

5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…

If A is NP-complete, and A has a polynomial time algorithm, then there is a polynomial time algorithm to find the largest clique in a directed. A clique in a graph G(V, E) is a subset of vertices V’ such that there is an edge (u,v) in E between every two vertices u and v in V’. For example, the four vertices u2, u3,u5, and u6 form the largest clique in the graph below.