Let $G_X = (X, \mathcal{E}_X)$ and $G_Y = (Y, \mathcal{E}_Y)$ each be a D778: Graph.
An D548: Ordered pair $G_Z = (Z, \mathcal{E}_Z)$ is an intersection of $G_X$ and $G_Y$ if and only if

(1)	$Z = X \cap Y$ (D76: Set intersection)
(2)	$\mathcal{E}_Z = \mathcal{E}_X \cap \mathcal{E}_Y$ (D76: Set intersection)