The structure of total dominating sets

Recently, Bacso and Tuza proved a theorem that gives a full characterization of the graphs that hereditarily have a connected dominating set satisfying prescribed hereditary properties. Their result is a 'high point' in the development of structural domination and completely solves a question which was (implicitely) stated 25 years ago.

Using their result, we derive a characterization of the graphs that hereditarily have a total dominating set whose connected components satisfy certain prescribed hereditary properties. This is the total domination equivalent to the theorem of Bacso and Tuza. In particular, our theory provides a characterization of the graphs that hereditarily have a total dominating set inducing the disjoint union of complete graphs. This inherits a characterization of the graphs whose any subgraph has a vertex-dominating induced matching. However, some cases do not permit a 'nice' characterization. We also discuss this phenomenon and give some partial characterizations.