  We enumerate certain geometric equivalence classes of subgraphs induced by
Hamiltonian paths and cycles in complete graphs. Said classes are orbits under
the action of certain direct products of dihedral and cyclic groups on sets of
strings representing subgraphs. These orbits are enumerated using Burnside's
lemma. The technique used also provides an alternative proof of the formulae
found by S. W. Golomb and L. R. Welch which give the number of distinct
$n$-gons on fixed, regularly spaced vertices up to rotation and optionally
reflection.
