NP-Completeness reduction, using a same input We have problem X and Y that we know is NP-Complete. Problem X uses graph G as an input and Problem Y uses graph G and constant k as an input.

Problem we are trying to reduce to, which I will call Z, is reasoning whether given a graph G and constant k, does G have X or Y.

I have been trying to solve this problem and an idea I came up with is to use an input G and k which can serve as an input for both X and Y, and reduce it to G' and k' where G=G' and k=k' for problem Z

Is it a valid thing to reduce without changing an input?
You can do whatever you want as long as you satisfy the definitions.

For example, a polynomial time reduction from a language $A$ to a language $B$ is a function $f$ satisfying the following two properties:


$f$ can be computed in polynomial time.
For every $x$, we have $x \in A$ iff $f(x) \in B$.


Any function $f$ satisfying these conditions is a polynomial time reduction from $A$ to $B$.

In particular, if $A$ is NP-hard and there exists a function $f$ satisfying these two conditions, then $B$ is also NP-hard.

The definition doesn't specify anything beyond these two properties. In mathematics nothing is left unsaid; everything which is not forbidden is actually allowed. All the cards are open. The rules of the game are public, and there is no need to ask a judge about any of them.