A technical clarification on subgraph isomorhism Let $G$ be a graph on $|\mathcal{V}(G)|$ vertices. If $H$ is another graph that contains a clique of size $|\mathcal{V}(G)|$, then does it mean $G$ is subgraph isomorphic to $H$? Does this mean that all graphs with vertices less than $|\mathcal{V}(G)|$ are subgraph isomorphic to $H$?

If the above is not the case, is there another term for the exact presence of $G$ in $H$ as a structure preserved subgraph (when I say structure preserved, that means, edges are mapped to edges and non-edges are mapped to non-edges in the subgraph of $H$)?
The subgraph isomorphism problem is "Is $G$ isomorphic to some subgraph of&nbsp;$H$?" so, yes, if $H$ contains a $k$-clique, that clique on its own has subgraphs isomorphic to any graph on $k$&nbsp;or fewer vertices. Your "exact presence" version is the induced subgraph isomorphism problem: is $G$ isomorphic to some induced subgraph of&nbsp;$H$? Both versions are NP-complete.

You could have found this trivially by Googling for "subgraph isomorphism problem".