# Asymptotic bounds on total domination in regular graphs

## TL;DR

This paper establishes new upper bounds on the minimum total dominating set size in regular graphs, leveraging local algorithms and probabilistic methods for graphs with large girth.

## Contribution

It introduces novel upper bounds for total domination in regular graphs using local algorithm analysis, applicable to random and large-girth graphs.

## Key findings

- New upper bounds for total domination in regular graphs
- Bounds are effective for graphs with large girth
- Analysis based on local algorithms and probabilistic methods

## Abstract

We find new upper bounds on the size of a minimum totally dominating set for random regular graphs and for regular graphs with large girth. These bounds are obtained through the analysis of a local algorithm using a method due to Hoppen and Wormald [Local algorithms, regular graphs of large girth, and random regular graphs. Combinatorica 38(3) (2018), 619-664].