On the existence of total dominating subgraphs with a prescribed additive hereditary property

Recently, Bacsó and Tuza gave a full characterization of the graphs for which every connected induced subgraph has a connected dominating subgraph satisfying an arbitrary prescribed hereditary property. Using their result, we derive a similar characterization of the graphs for which any isolate-free induced subgraph has a total dominating subgraph that satisfies a prescribed additive hereditary property. In particular, we give a characterization for the case where the total dominating subgraphs are a disjoint union of complete graphs. This yields a characterization of the graphs for which every isolate-free induced subgraph has a vertex-dominating induced matching, a so-called induced paired-dominating set.

► Characterization of graphs having total dominating subgraph in given graph class. ► Characterization of graphs with total dominating set forming disjoint cliques. ► Characterization of graphs having an induced paired-dominating set.