  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.
