Question: The Longest Cycle problem (LONG-CYCLE) accepts a vertex v in graph G, along with a length k, and returns whether the graph contains a cycle

The Longest Cycle problem (LONG-CYCLE) accepts a vertex v in graph G, along with a length k, and returns whether the graph contains

The Longest Cycle problem (LONG-CYCLE) accepts a vertex v in graph G, along with a length k, and returns whether the graph contains a cycle of length k or more that includes v The Longest Path problem (LONG-PATH), which we have discussed in class, accepts two vertices u and v in a graph G and a distance k, and it returns whether G contains a path of length k or longer between u and v 1. If LONG-CYCLE E NP-Hard and LONG-PATHE NP, which of the reduc- tions below would be used to prove that LONG-PATHE NP-Complete? You may assume that both reductions are correct. Input: G (V, E): graph with n vertices and m edges Input: n, m: order and size of G vertex Input of G Input k: cycle length Output: LONG-CYCLEG, v, k whether a is part of a cycle of length k or greater in G 1 Algorithm: ReductionOne 2 for u EN(v) do 3 G. emove Edge(u, v); 4 if LONG PATH (G, u, v, k -1) then return true; 6 end G.Ad Edge(u, v); 8 end 9 return false

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