Definition:Induced Subgraph

Let $$G = \left({V \left({G}\right), E \left({G}\right)}\right)$$ be a graph.

Let $$V' \subseteq V$$ be a subset of vertices of $$G$$.

The subgraph of $$G$$ induced by $$V'$$ is the subgraph $$G' = \left({V' \left({G'}\right), E' \left({G'}\right)}\right)$$ of $$G$$ that:
 * $$G'$$ has the vertex set $$V'$$;


 * For all $$u, v \in V'$$, $$e = uv \in E \left({G}\right)$$ iff $$e \in E' \left({G'}\right)$$.

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.