  Given a graph G, we construct a simple, convex polytope whose face poset is
based on the connected subgraphs of G. This provides a natural generalization
of the Stasheff associahedron and the Bott-Taubes cyclohedron. Moreover, we
show that for any simplicial Coxeter system, the minimal blow-ups of its
associated Coxeter complex has a tiling by graph-associahedra. The geometric
and combinatorial properties of the complex as well as of the polyhedra are
given. These spaces are natural generalizations of the Deligne-Knudsen-Mumford
compactification of the real moduli space of curves.
