Total Domination in Regular Graphs

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 [Hoppen, C., and N. C. Wormald. Local algorithms, regular graphs of large girth, and random regular graphs. Combinatorica 38(3) (2018), 619-664.].