  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.
