\begin{definition}[Definition:Isomorphism (Graph Theory)]
Let $G = \struct {\map V G, \map E G}$ and $H = \struct {\map V H, \map E H}$ be graphs.
Let there exist a bijection $F: \map V G \to \map V H$ such that for each edge $\set {u, v} \in \map E G$, there is an edge $\set {\map F u, \map F v} \in \map E H$.
That is, that:
:$F: \map V G \to \map V H$ is a homomorphism, and
:$F^{-1}: \map V H \to \map V G$ is a homomorphism.
Then $G$ and $H$ are '''isomorphic''', and this is denoted $G \cong H$.
The function $F$ is called an '''isomorphism''' from $G$ to $H$.
\end{definition}