# Definition:Isomorphism (Graph Theory)

## Definition

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 $H$.

## Linguistic Note

The word isomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus isomorphism means equal structure.