For each decision problem D below, state whether D e P or D E NP, then justify your claim For decision problems in P, describe an algorithm that decides the problem in polytime (including a brief argument that your decider is correct and runs in polytime). For decision problems in NP, describe an algorithm that verifies the problem in polytime (in- cluding a brief argument that your verifier is correct and runs in polytime), and give a detailed reduction to show that the decision problem is NP-hard- for your reduction(s), you must use one of the problems shown to be NP-hard during lectures or tutorials. For each of the following decision problems, recall that a cycle an an undirected graph is simple if it contains no repeated vertex or edge. (a) [8 marks] ExacTCYCLE Input: An undirected graph G and a non-negative integer k. Question: Does G contain some simple cycle on exactly k vertices? (b) [8 marks] SMALLCYCLE Input: An undirected graph G and a non-negative integer k. Question: Does G contain some simple cycle on at most k vertices? (c) [8 marks] LARGECYCLE Input: An undirected graph G and a non-negative integer k. Question: Does G contain some simple cycle on at least k vertices