For any finite set A of n points in general position in R^2, we define a
(3n-3)-dimensional simple polyhedron whose face poset is isomorphic to the
poset of "non-crossing marked graphs" with vertex set A, where a marked graph
is defined as a geometric graph together with a subset of its pointed vertices.
The poset of non-crossing graphs on A appears as the complement of
the star of a face in that polyhedron.