Bi-Symmetric Polyhedral Cages with Maximally Connected Faces and Small Holes

Polyhedral cages (p-cages) describe the geometry of some families of artificial protein cages. We identify the p-cages made out of families of equivalent polygonal faces such that the faces of one family have five neighbors and $P_1$ edges, while those of the other family have six neighbors and $P_2$ edges. We restrict ourselves to polyhedral cages where the holes are adjacent to four faces at most. We characterize all p-cages with a deformation of the faces, compared to regular polygons, not exceeding 10%.