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Our work studies the enumeration and random generation of unlabeledcombinatorial classes of unrooted graphs. While the technique of vertexpointing provides a straightforward procedure for analyzing a labeled class ofunrooted graphs by first studying its rooted counterpart, the existence ofnontrivial symmetries in the unlabeled case causes this technique to breakdown. Instead, techniques such as the dissymmetry theorem (of Otter) and cyclepointing (of Bodirsky et al.) have emerged in the unlabeled case, with theformer providing an enumeration of the class and the latter providing both anenumeration and an unbiased sampler. In this work, we extend the power of thedissymmetry theorem by showing that it in fact provides a Boltzmann sampler forthe class in question. We then present an exposition of the cycle pointingtechnique, with a focus on the enumeration and random generation of theunderlying unpointed class. Finally, we apply cycle pointing to enumerate andimplement samplers for the classes of distance-hereditary graphs and three-leafpower graphs.

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