Answer in Python only You are given an undirected graph G = (V, E) containing N nodes and M edges. The nodes are numbered from 1 to N. A subset C of V is a vertex cover if for every edge (u, v) E E, at least one of u and v belong to C. Note that C = V is always a vertex cover. Consider a partition of V into two sets A and B. It is said to be a valid partition, if the following two conditions are satisfied: A should be a vertex cover. And for each i such that 1 ≤ i ≤n/2, nodes 2*i and 2*i- 1 don't belong to the same set (i.e. one belongs to set A and the other to set B). Determine if a valid partition exists. If it exists, provide an example of one valid partition. Input 1 32 12 23 Output

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…