Generation of simple quadrangulations of the sphere Original Research Article

class can be generated by starting with some basic graphs in the class and applying a sequence of local modifications. The duals of our algorithms generate classes of quartic (4-regular) planar graphs. In the case of minimum degree 3, our result is a strengthening of a theorem of Nakamoto and almost implicit in Nakamotoʹs proof. In the case of 3-connectivity, a corollary of our theorem matches a theorem of Batagelj. However, Batageljʹs proof contained a serious error which cannot easily be corrected. We also give a theoretical enumeration of rooted planar quadrangulations of minimum degree 3, and some counts obtained by a program of Brinkmann and McKay that implements our algorithm