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

• Published in: Journal of algebraic combinatorics Vol. 54; no. 4; pp. 1119 - 1135
• Main Authors: Gross, Jonathan L., Mansour, Toufik, Tucker, Thomas W.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2021; Springer Nature B.V
• Subjects: Algorithms; Combinatorics; Computer Science; Convex and Discrete Geometry; Enumeration; Graph theory; Graphs; Group Theory and Generalizations; Hypercubes; Lattices; Mathematics; Mathematics and Statistics; Operators (mathematics); Order; Ordered Algebraic Structures; Polynomials; Sequences; Set theory; Rotation system; Gray code; Partial duality; Primary: 05C10; Ribbon graph; Poincaré dual; Genus polynomial; Petrie dual; Euler-genus
• ISSN: 0925-9899; 1572-9192; 1572-9192
• DOI: 10.1007/s10801-021-01040-y

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.