# Euler-genus distributions of cubic Halin graphs

## TL;DR

This paper develops a recurrence relation and explicit formulas for the Euler-genus distribution of cubic caterpillar-Halin graphs, advancing understanding of their topological embeddings.

## Contribution

It introduces a recurrence relation for the Euler-genus polynomial of cubic caterpillar-Halin graphs and provides explicit formulas for embeddings into surfaces with low Euler-genus.

## Key findings

- Recurrence relation for Euler-genus polynomial derived
- Explicit formulas for embeddings into surfaces with Euler-genus 0, 1, 2 obtained
- Advances in calculating topological embeddings of cubic Halin graphs

## Abstract

Gross derived an $O(n^2)$-time algorithm to calculate the genus distribution of a given cubic Halin graph. In this paper, with the help of overlap matrix, we get a recurrence relation for the Euler-genus polynomial of cubic caterpillar-Halin graphs. Explicit formulas for the embeddings of cubic caterpillar-Halin graph into a surface with Euler-genus 0, 1 and 2 are also obtained.