Given an undirected graph, I define a structure called k-key as a path containing $k$ vertices which are connected to a simple cycle which contains $k$ vertices as well.
Here's the k-key problem: given an undirected graph $G$ and a number $k$, decide whether $G$ contains k $k$-key.
I want to show that the k-key problem is a NP-complete.
I want to make a reduction from the 'Undirected Hamiltonian Cycle' problem in which the input is a graph, and the problem is to decide whether it contains a Hamiltonian path. I already know that this problem is NP-complete. The input for the reduction would be an undirected graph $G$ and the output is $G'$ graph and $k$. Can you please help me understand what manipulation I should do to the original graph in order to show this reduction? And why should it work?