Recursive generation of simple planar 5-regular graphs and pentangulations

We describe how the 5-regular simple planar graphs can all be obtained from an elementary family of starting graphs by repeatedly applying a few local expansion operations. The proof uses an amalgam of theory and computation. By incorporating the recursion into the canonical construction path method of isomorph rejection, a generator of non-isomorphic embedded 5-regular planar graphs is obtained with time complexity O(n) per isomorphism class. A similar result is obtained for simple planar pentangulations with minimum degree 2.