Definition:Subgraph

Definition
A graph $H = \left({V \left({H}\right), E \left({H}\right)}\right)$ is called a subgraph of a graph $G = \left({V \left({G}\right), E \left({G}\right)}\right)$ if $V \left({H}\right)\subseteq V \left({G}\right)$ and $E \left({H}\right)\subseteq E \left({G}\right)$.

That is to say, it contains no vertices or edges that are not in the original graph.

If $H$ is a subgraph of $G$, then:
 * $G$ contains $H$;
 * $H$ is contained in $G$