My understanding is that when reducing one problem known to be NPC (e.g., HAM-Cycle) to another problem known to be NP (e.g., Ham-Path) we have shown that the second problem (the target of the reduction, or HAM-Path in the example) is NPC.
Now I also understand that this reduction must be many-one (or, equivalently, a Karp reduction.)
I came a across a reduction from Ham-Cycle to Ham-Path that called a decider for Ham-Path multiple times in the reduction algorithm. It seems that this would qualify as a Turing reduction and not a Karp reduction and so wouldn't qualify as a valid NPC reduction.
Reduction from Hamiltonian cycle to Hamiltonian path. I'm referring to the upvoted solution of Aryabhata (I don't have enough reputation to comment on his post.)
In general, I'm a little confused about the different kinds of reductions and which are valid for an NPC reduction (e.g., so clearly many-one and one-to-one reductions must both be valid for NPC proofs, right?)
Background: CLRS appears to define the reductions as mapping reductions and is the same as Sipser's book and because of the linked post this caused confusion. Upon further research on stack exchange, most things claim we need a many-one reduction, but were not sufficiently clear about what is going on.