Let $G_X = (X, \mathcal{E}_X)$ and $G_Y = (Y, \mathcal{E}_Y)$ each be a D778: Graph.
A D18: Map $f : X \to Y$ is a graph isomorphism with respect to $G_X$ and $G_Y$ if and only if

(1)	$f$ is a D4548: Graph homomorphism with respect to $G_X$ and $G_Y$
(2)	$f$ is an D976: Invertible map with an D216: Inverse map $f^{-1} : Y \to X$
(3)	$f^{-1}$ is a D4548: Graph homomorphism with respect to $G_Y$ and $G_X$