Growing by unfolding the 3-connected planar quadrangulations

Building on work by Batagelj, Brinkmann et al. proved that the 3-connected planar quadrangulations (3CPQ's) can be derived from the pseudo-double wheels, W_2k for k >= 3, via two local expansion operations P₁ and P₃, illustated above. An example of a pseudo-double wheel, W₁₄, is illustrated below.

P₁ is basically a surgery on the surface. By slitting edges and vertices along any path of three vertices, a quad-shaped wound opens that can be healed with new surface. Applied without restriction, this operation can generate any quadrangulation, but Brinkmann et al. enforce restrictions that make it possible to realize P₁ by unfolding triangulated quadrangles. P₃ can also be realized by a twist unfolding of triangulated quads, popping a 2-sided square into a cube in a single move. Here is an example that starts from a square (2 faces, and not 3-connected) and ends with 16 faces, and 3-connected.