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.

Chmutov使用功能区图在几何上定义了对任意边缘子集上的嵌入式图G进行部分庞加莱对偶化的方法。该系列论文的第一部分介绍了偏二项式多项式，该多项式根据图的Euler属属枚举了图G的所有可能的偏对偶，并且根据对偶化的边缘子集的选择可能会发生变化。Ellis-Monaghan和Moffatt扩展了部分对偶概念，使其包括Petrie对偶，Wilson对偶和两个对等运算符。艾布拉姆斯和埃利斯-莫纳汉已经给出了五个营集体的名义twualities。这一系列论文的第二部分推导了部分二次多项式的公式，这些公式对应于嵌入图的几个基本序列。在第三部分中，我们介绍了一种算法，用于为所有二次计算带状图G的部分二次多项式 ，其中包括将G的边缘子集组织成一个超立方体，并通过格雷代码遍历该超立方体。