Combinatorially equivalent polyhedra?

Question

What does it mean for two polyhedra to be combinatorially equivalent? I've looked on the internet but in vain. If it's not a standard definition, then it might help to say that I found this term in a paper on flattening polyhedra: Continuous flattening of convex polyhedra by Jin-ichi Itoh et al.

Answer 1

The set of all faces of a polytope, partially ordered by inclusion, is called its face lattice. Two polytopes are combinatorially equivalent if they have isomorphic face lattices.

A nice introduction to the combinatorial aspects of polytopes can be found in this pdf.

An equivalent condition for combinatorial equivalence between two polytopes is that their face-vertex incidence matrices differ only by column and row permutations.

Answer 2

For convex 3-dimensional polyhedra you can determine if two such polyhedra are combinatorially equivalent by looking at their vertex-edge graphs. It is known that a graph is the vertex-edge graph of a convex 3-dimensional polyhedron if and only if the graph is planar and 3-connected (Steinitz's Theorem). So if two convex 3-dimensional polyhedra have graphs which are isomorphic they are combinatorially equivalent. For higher dimensional convex polytopes you have to look at the face lattices.