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A total dominating set $S\subset V(G)$ is a set of vertices such that $\forall v\in V(G)$, $v$ has a neighbour in $S$. The minimum total dominating set of $G$ is a total dominating set of $G$ of minimum possible size. The minimum total domination number of $G$ is represented by $\gamma_t(G)$.It is known that MTDS is $NP-complete$ for general graphs, bipartite graphs, chordal graphs, etc. [REFERENCE: Henning, Michael A.. “A survey of selected recent results on total domination in graphs.” Discret. Math. 309 (2009): 32-63.]

Do we know anything about hardness of MTDS on triangle-free graphs of bounded degree?Or triangle-free graphs in general? ( A graph is triangle-free if any three vertices does not form a triangle).

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    $\begingroup$ All bipartites are triangle free. That answers your second question. $\endgroup$ Jul 3, 2023 at 9:23
  • $\begingroup$ Yes that's correct. $\endgroup$ Jul 3, 2023 at 9:32
  • $\begingroup$ It would be good if you could tell the motivation for studying this version since there could be many such variants. $\endgroup$ Jul 3, 2023 at 10:05
  • $\begingroup$ So I am studying this specific kind of proper coloring of graphs called "cd-coloring" in which every color class is dominated by atleast one vertex. The cd-chromatic number is denoted by $\chi_{cd}$. So it is proven that for triangle -free graphs $G$ ,$\chi_{cd}(G)=\gamma_t(G)$. Right now my motive is to study cd-colorability in regular graphs(in particular cubic graphs). I want to prove this is $NP-hard$ for triangle-free cubic graphs. So I tried to reduce *total domination in cubic graphs*(which is $NP-complete$) to cd -colorability in cubic graphs, but till now am not successful. $\endgroup$ Jul 3, 2023 at 13:22
  • $\begingroup$ (Continuing) So I am searching for total-domination $NP-completeness$ results for triangle-free graphs of bounded degree, So that I can reduce it to cd-colorability of some triangle-free $k$-regular graphs, thereby proving it $NP-complete$. $\endgroup$ Jul 3, 2023 at 13:25

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The problem is indeed $\mathsf{NP}$-hard on bipartite graphs (which are triangle-tree) of degree at most $3$. See Theorem 6 of this paper.

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