  A graph associahedron is a polytope dual to a simplicial complex whose
elements are induced connected subgraphs called tubes. Graph associahedra
generalize permutahedra, associahedra, and cyclohedra, and therefore are of
great interest to those who study Coxeter combinatorics.
  This thesis characterizes nested complexes of simplicial complexes, which we
call $\Delta$-nested complexes. From here, we can define P-nestohedra by
truncating simple polyhedra, and in more specificity define P-graph
associahedra, which are realized by repeated truncation of faces of simple
polyhedra in accordance with tubes of graphs.
  We then define hypercube-graph associahedra as a special case.
Hypercube-graph associahedra are defined by tubes and tubings on a graph with a
matching of dashed edges, with tubes and tubings avoiding those dashed edges.
These simple rules make hypercube-graph tubings a simple and intuitive
extension of classical graph tubings. We explore properties of $\Delta$-nested
complexes and P-nestohedra, and use these results to explore properties of
hypercube-graph associahedra, including their facets and faces, as well as
their normal fans and Minkowski sum decompositions. We use these properties to
develop general methods of enumerating $f$-polynomials of families of
hypercube-graph associahedra. Several of these hypercube-graphs correspond to
previously-studied polyhedra, such as cubeahedra, the halohedron, the type
$A_n$ linear $c$-cluster associahedron, and the type $A_n$ linear $c$-cluster
biassociahedron. We provide enumerations for these polyhedra and others.
