# Recognizing hyperelliptic graphs in polynomial time

## TL;DR

This paper introduces an efficient method to recognize hyperelliptic graphs, which are multigraphs of gonality 2, using reduction rules that operate in near-linear time, advancing understanding in graph theory and algorithms.

## Contribution

It provides a safe and complete set of reduction rules for identifying hyperelliptic graphs in polynomial time, specifically for three gonality variants.

## Key findings

- Recognition algorithms run in O(n log n + m) time.
- Hyperelliptic graphs can be characterized by specific reduction rules.
- The approach advances graph recognition techniques for special multigraph classes.

## Abstract

Recently, a new set of multigraph parameters was defined, called "gonalities". Gonality bears some similarity to treewidth, and is a relevant graph parameter for problems in number theory and multigraph algorithms. Multigraphs of gonality 1 are trees. We consider so-called "hyperelliptic graphs" (multigraphs of gonality 2) and provide a safe and complete sets of reduction rules for such multigraphs, showing that for three of the flavors of gonality, we can recognize hyperelliptic graphs in O(n log n+m) time, where n is the number of vertices and m the number of edges of the multigraph.