Given a graph Γ , we construct a simple, convex polytope, dubbed graph-associahedra, whose face poset is based on the connected subgraphs of Γ . 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.

This paper is built on the following observation: the purity of the mixed Hodge structure on the cohomology of Brown's moduli spaces is essentially equivalent to the freeness of the dihedral operad underlying the gravity operad. We prove these two facts by relying on both the geometric and the algebraic aspects of the problem: the complete geometric description of the cohomology of Brown's moduli spaces and the coradical filtration of cofree cooperads. This gives a conceptual proof of an identity of Bergström-Brown which expresses the Betti numbers of Brown's moduli spaces via the inversion of a generating series. This also generalizes the Salvatore-Tauraso theorem on the nonsymmetric Lie operad.

Given any finite graph G, we offer a simple realization of the graph-associahedron PG using integer coordinates.2000 Mathematics Subject Classification. Primary 52B11.

Given a graph G, we construct a convex polytope whose face poset is based on marked subgraphs of G. Dubbed the graph multiplihedron, we provide a realization using integer coordinates. Not only does this yield a natural generalization of the multiphihedron, but features of this polytope appear in works related to quilted disks, bordered Riemann surfaces, and operadic structures. Certain examples of graph multiplihedra are related to Minkowski sums of simplices and cubes and others to the permutohedron.

The polytope structure of the associahedron is decomposed into two categories, types and classes. The classification of types is related to integer partitions, whereas the classes present a new combinatorial problem. We solve this and incorporate the results into properties of the real moduli space of Riemann spheres. Connections are discussed with relation to classic combinatorial problems as well as to other sciences. The Real Moduli Space MotivationThe Riemann moduli space M n g of surfaces of genus g with n marked points has become a central object in mathematical physics. Introduced in Algebraic Geometry, there is a natural compactification M n g of these spaces; their importance was emphasized by Grothendieck in his famous Esquisse d'un programme [12]. The special case when g = 0, the space M n 0 (C) of n punctures on the sphere CP 1 , is a building block leading to higher genera. They play a crucial role in the theory of Gromov-Witten invariants and symplectic geometry. Furthermore, they appear in the work of Kontsevich on quantum cohomology, and are closely related to the operads of homotopy theory. We look at the real points M n 0 (R) of this space; these are the set of fixed points under complex conjugation. This moduli space will provide the motivation to study certain combinatorial objects. Definition 1.1.1. The real Deligne-Knudsen-Mumford moduli space M n 0 (R) is a compactification of the configurations of n labeled points on RP 1 quotiented by the action of PGl 2 (R).M n 0 (R) is a manifold without boundary of real dimension n − 3. The moduli space is a point when n = 3 and RP 1 when n = 4. For n > 4, these spaces become nonorientable. Figure 1 shows M 5 0 (R) (shaded) as the connected sum of five copies of RP 2 , 71