Abstract
We obtain new upper bounds on the size of a minimum total dominating set for random regular graphs and for regular graphs with large girth. In particular, they imply that an upper bound conjectured by Thomassé and Yeo (2007) holds asymptotically almost surely for 5-regular graphs and holds for all 5-regular graphs with sufficiently large girth. Our bounds are obtained through the analysis of a local algorithm using a method due to Hoppen and Wormald (2018).
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