I have been studying NP-Complete problems and I saw the Dense Subgraph problem. Then I saw that they are trying to show that the problem is NP (see below quote), but I can't understand how it verifies that time is polynomial.
Problem: Dense Subgraph
Input: A graph $G$, and integers $k$ and $y$.
Output: Does $G$ contain a subgraph with exactly $k$ vertices and at least $y$ edges?
To prove Dense Subgraph is NP-complete, we show that Dense Subgraph is in NP and is NP-hard by reducing CLIQUE problem to Dense Subgraph.
Dense Subgraph is NP: the polynomial time verifier will take $(G,k,y)$ and $H = (V', E')$ as certificate and check if $H$ is a subgraph of $G$, $|V'| = k$, $|E'| \geq y$.