I was trying to understand the Wikipedia proof for NP-completeness of subgraph isomorphism by reduction from the clique problem. It's really just one sentence:
Let $H$ be the complete graph $K_k$; then, the answer to the subgraph isomorphism problem for $G$ and $H$ is equal to the answer to the clique problem for $G$ and $k$.
In the subgraph isomorphism problem, $G$ and $H$ are not constrained in any way (e.g., $H$ is not constrained to complete graphs). Yet, in the proof, Wikipedia just states "let $H$ be the complete graph $K_k$". It's not defined what $K_k$ represents or why $H$ suddenly becomes constrained to a complete graph.