  To any graph $G$ one can associate a toric variety $X(\mathcal{P}G)$,
obtained as a blowup of projective space along coordinate subspaces
corresponding to connected subgraphs of $G$. The polytope of this toric variety
is the graph associahedron of $G$, a class of polytopes that includes the
permutohedron, associahedron, and stellahedron. We show that the space
$X(\mathcal{P}{G})$ is isomorphic to a Hassett compactification of $M_{0,n}$
precisely when $G$ is an iterated cone over a discrete set. This may be viewed
as a generalization of the well-known fact that the Losev--Manin moduli space
is isomorphic to the toric variety associated to the permutohedron.
