\begin{definition}[Definition:Induced Subgraph]
Let $G = \struct {\map V G, \map E G}$ be a graph.
Let $V' \subseteq V$ be a subset of vertices of $G$.
The '''subgraph of $G$ induced by $V'$''' is the subgraph $G' = \struct {\map {V'} {G'}, \map {E'} {G'} }$ of $G$ such that:
:$G'$ has the vertex set $V'$
:For all $u, v \in V'$, $e = u v \in \map E G$ {{iff}} $e \in \map {E'} {G'}$.
That is, it contains all the edges of $G$ that connect elements of the given subset of the vertex set of $G$, and only those edges.
Category:Definitions/Graph Theory
\end{definition}