# Methods of Enumerating Two Vertex Maps of Arbitrary Genus

## TL;DR

This paper offers an alternative proof and simplification of the Goulden-Slofstra formula for counting two-vertex maps by genus, extending the Harer-Zagier formula related to moduli space Euler characteristics.

## Contribution

It introduces a new proof and simplification of the Goulden-Slofstra formula, facilitating future generalizations to maps with more vertices.

## Key findings

- Provided an alternate proof of the Goulden-Slofstra formula
- Simplified the existing formula for two-vertex maps
- Laid groundwork for generalization to maps with more vertices

## 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.