The geometric combinatorics of pseudographs and beyond
Dr. Stefan Forcey, Dr M. Carr (Brandeis) and Dr. S. Devadoss (Williams)
Given a simple graph G, the graph associahedron KG is a simple polytope whose face poset is based on the connected subgraphs of G. We define and construct graph associahedra in a general context, for pseudographs with loops and multiple edges, which are also allowed to be disconnected. We then consider deformations of pseudograph associahedra as their underlying graphs are altered by edge contractions and edge deletions. Open research questions include the enumeration of vertices and faces of the polytopes, as well as investigation of the conjectured CW-complex associahedra. Even more conjectural is the existence of colored versions of these polytopes, called pseudograph multiplihedra. Here is our paper on pseudograph associahedra, and here is the paper on graph multiplihedra. For more information see this page of current research projects.