Reduction between parametrized problems
You are asking two questions, I will answer the second. Parameterized reduction from problem $P$ to $Q$ is a function $\phi$ with the following properties : 


$\phi(x)$ can be computed in time  $f (k)·|x|^{O(1)}$, where $k$ is the parameter of $x$,
$\phi(x)$ is a yes-instance of $Q$ if and only if $x$ is a yes-instance of $P$.
If $k$ is the parameter of $x$ and $k′$ is the parameter of $\phi(x)$, then $k′ ≤ g(k)$ for some function $g$.



  Note that $f(k)$ need not to be a polynomial function in $k$, it could be like $2^k$ or $c^k$, where $c$ is constant.


Example: INDEPENDENT-SET instance $(G,K)$ into CLIQUE instance $(G',K)$ is a parameterized reduction. You can verify all three conditions mentioned above.

Reference : http://www.cs.bme.hu/~dmarx/papers/marx-warsaw-fpt3