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

TL;DR

This paper presents an optimal algorithm for exhaustively generating rooted trivalent diagrams, ensuring non-redundant, isomorphism-free listings with bounded amortized time, and provides a theoretical framework for similar combinatorial generation tasks.

Contribution

The paper introduces a new, optimal algorithm for generating rooted trivalent diagrams with the CAT property, and offers a reusable theoretical framework for combinatorial structure enumeration.

Findings

Algorithm has optimal amortized performance (CAT property).

Generates non-redundant, isomorphism-free diagrams.

Framework applicable to other complex combinatorial structures.

Abstract

A trivalent diagram is a connected, two-colored bipartite graph (parallel edges allowed but not loops) such that every black vertex is of degree 1 or 3 and every white vertex is of degree 1 or 2, with a cyclic order imposed on every set of edges incident to to a same vertex. A rooted trivalent diagram is a trivalent diagram with a distinguished edge, its root. We shall describe and analyze an algorithm giving an exhaustive list of rooted trivalent diagrams of a given size (number of edges), the list being non-redundant in that no two diagrams of the list are isomorphic. The algorithm will be shown to have optimal performance in that the time necessary to generate a diagram will be seen to be bounded in the amortized sense, the bound being independent of the size of the diagrams. That's what we call the CAT property. 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.