We say that $G' = (V',E')$ is a subgraph of $G = (V,E)$ if $V' \subseteq V$ and $E' \subseteq E$ In this case we write $G' \subseteq G$

If $G'$ contains all edges of $G$ that join two vertices in $V'$ then $G'$ is said to be the subgraph induced or spanned by $V'$ and is denoted by $G[V']$ Thus, a subgraph $G'$ of $G$ is an induced subgraph if $G' = G[V(G')]$ If $V' = V$ then $G'$ is said to be a spanning subgraph of $G$

Often, new graphs are constructed from old ones by deleting or adding some vertices and edges. If $W \subset V(G)$ then $G - W = G[V\setminus W]$ is the subgraph of $G$ obtained by deleting the vertices in $W$ and all edges incident with them. Similarly, if $E' \subseteq E(G)$ then $G - E' = (V(G),E(G) \setminus E')$ If $W = {w}$ and $E' = {xy}$ then this notation is simplified to $G - w$ and $G - xy$ Similarly, if $x$ and $y$ are nonadjacent vertices of $G$ then $G + xy$ is obtained from $G$ by joining $x$ to $y$

Adapted with permission of the author from Modern Graph Theory by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.