Assume a weighted graph G and a positive value k are given, and you are looking for the complexity of finding a cycle with total weight k; beside you know that no Hamiltonian cycle in G has total weight k. (G is an arbitrary weighted graph, beside you know no hamiltonian cycle would satisfy the total weight k)
1-can someone use reduction from knapsack to conclude that the complexity of finding a cycle with total weight k In G is NP-com? (I guess no!)
2-Is there any idea about the complexity of finding a cycle in G with total weight k when you know no Hamiltonian cycle in G convinces total weight k?