Given a non-Hausdorff space $X$, one can form a graph $G_X$: vertices the points of $X$, edges indicating point pairs not separated by open sets. Up to graph-theoretically (but not topologically) isolated points, I believe one can achieve any graph as a $G_X$ just by realizing edges with copies of the so-called bug-eyed line.
Hausdorffness can fail for the Wallman compactification $\gamma X$ of a Hausdorff, but not normal, space $X$. As such, do there exist graph-theoretic constrains on the non-isolated points of$ G_{\gamma X}$?
