Title: Methods of Enumerating Two Vertex Maps of Arbitrary Genus,
Abstract: This paper provides an alternate proof to parts of the Goulden-Slofstra
formula for enumerating two vertex maps by genus, which is an extension of the
famous Harer-Zagier formula that computes the Euler characteristic of the
moduli space of curves. This paper also shows a further simplification to the
Goulden-Slofstra formula. Portions of this alternate proof will be used in a
subsequent paper, where it forms a basis for a more general result that applies
for a certain class of maps with an arbitrary number of vertices.