Partial duality for ribbon graphs, III: a Gray code algorithm for enumeration

Partially Poincaré-dualizing an embedded graph G on an arbitrary subset of edges was defined geometrically by Chmutov, using ribbon graphs. Part I of this series of papers introduced the partial-duality polynomial, which enumerates all the possible partial duals of the graph G, according to their Euler-genus, which can change according to the selection of the edge subset on which to dualize. Ellis-Monaghan and Moffatt have expanded the partial-duality concept to include the Petrie dual, the Wilson dual, and the two triality operators. Abrams and Ellis-Monaghan have given the five operators the collective name twualities. Part II of this series of papers derived formulas for partial-twuality polynomials corresponding to several fundamental sequences of embedded graphs. Here in Part III, we present an algorithm to calculate the partial-twuality polynomial of a ribbon graph G, for all twualities, which involves organizing the edge subsets of G into a hypercube and traversing that hypercube via a Gray code.