Enumeration and Random Generation of Unlabeled Classes of Graphs: A Practical Study of Cycle Pointing and the Dissymmetry Theorem

TL;DR

This paper explores advanced combinatorial techniques for enumerating and randomly generating unlabeled unrooted graphs, extending the dissymmetry theorem to provide Boltzmann samplers and applying cycle pointing to specific graph classes.

Contribution

It extends the dissymmetry theorem to include Boltzmann samplers and provides a detailed exposition of cycle pointing for unrooted graph classes.

Findings

Dissymmetry theorem can be used to create Boltzmann samplers for unlabeled graphs.

Cycle pointing effectively enumerates and samples unpointed graph classes.

Applied techniques successfully to distance-hereditary and three-leaf power graphs.

Abstract

Our work studies the enumeration and random generation of unlabeled combinatorial classes of unrooted graphs. While the technique of vertex pointing provides a straightforward procedure for analyzing a labeled class of unrooted graphs by first studying its rooted counterpart, the existence of nontrivial symmetries in the unlabeled case causes this technique to break down. Instead, techniques such as the dissymmetry theorem (of Otter) and cycle pointing (of Bodirsky et al.) have emerged in the unlabeled case, with the former providing an enumeration of the class and the latter providing both an enumeration and an unbiased sampler. In this work, we extend the power of the dissymmetry theorem by showing that it in fact provides a Boltzmann sampler for the class in question. We then present an exposition of the cycle pointing technique, with a focus on the enumeration and random generation of the underlying unpointed class. Finally, we apply cycle pointing to enumerate and implement samplers for the classes of distance-hereditary graphs and three-leaf power graphs.