A hamiltonian cycle is a cycle which crosses all of the vertices.
Problem (i): An arbitrary simple graph $G$ is given. Does $G$ have a hamiltonian cycle?
Assume that we know problem (i) is NP-Complete. How can we show that problem (ii) is NP-Complete, too?
Problem (ii): A simple graph $G$ is given. Does $G$ have a cycle of length $n/2$? ($n$ is the number of vertices)
Note 1: I know that i should use a reduction. But, The problem is that i don't know a way to translate the input of problem (ii) into an input of problem (i). Any idea?
Note 2: I want to learn a reduction. So, The thing that matters is the reduction, not just the answer of the question. I want a translation function which takes a simple graph as input and returns another simple graph which can be seen as the input of problem (i).
Thanks in advance.