The kernel of a parameterized problem $L$ is a reduction $(x,k) \mapsto (x',k')$ such that:
- $(x,k) \in L \Leftrightarrow (x',k') \in L$
- $|x'| \leq f(k)$ for some function $f$
- $k' \leq g(k)$ for some function $g$
- transformation must be computed in polynomial time.
Can I say that the reduction is a Karp reduction $(x,k) \mapsto (x',k')$ such that
- $(x,k) \in L \Leftrightarrow (x',k') \in L$
- $|x'| \leq f(k)$ for some function $f$
- $k' \leq g(k)$ for some function $g$
I am confused because the language is a binary language. Perhaps, this is a naive question, but I am not sure if I can call it Karp reduction. It seems that the Karp reduction is not used in parameterized complexity.