An Optimal Algorithm to Generate Pointed Trivalent Diagrams and Pointed Triangular Maps

A trivalent diagram is a connected graph with cyclic orientations and degree conditions on its vertices. It is the combinatorial description of an unembedded trivalent ribbon graph. We shall describe and analyze an algorithm giving an exhaustive list of trivalent diagrams of a given size. The list being nonredundant in that no two diagrams of the list are isomorphic. The algorithm will be shown to have optimal performances in that the time necessary to generate a diagram will be shown to be bounded in the amortized sense. What is striking is that the bound is uniform on the size of the diagrams being generated. One objective of the paper is to provide a reusable theoretical framework for algorithms generating exhaustive lists of complex combinatorial structures with attention paid to the case of unlabeled structures and to those generators having the CAT property.