The associahedron is a well-studied polytope. For n dimensions, its vertices are counted by the n-th Catalan number, a sequence starting 1,1,2,5,14,42,… and which counts many, many, many combinatorial objects, such as Dyck paths, planar binary trees, noncrossing set partitions, and polygonal triangulation. There is a well-known generalization of the associahedron, called the graph associahedron, which is obtained by truncating faces of a simplex which correspond to induced connected subgraphs. My research generalizes graph associahedra from simplices to all simple polytopes. Given any simple polytope P and a graph on the facets of that polytope, repeated truncations of P create a new polytope, called a P-graph associahedron. When P is a hypercube, these polytopes have combinatorial interpretations related to graphs on positive and negative vertices. We are continuing to examine the properties of these polytopes.

Organizer: Benjamin Hollering