A semitotal dominating set of a graph G with no isolated vertex is a dominating set D of G such that every vertex in D is within distance two of another vertex in D. The minimum size γt2(G) of a semitotal dominating set of G is squeezed between the domination number γ(G) and the total domination number γt(G).
Semitotal Dominating Set is the problem of finding, given a graph G, a semitotal dominating set of G of size γt2(G). In this paper, we continue the systematic study on the computational complexity of this problem when restricted to special graph classes. In particular, we show that it is solvable in polynomial time for the class of graphs of bounded mim-width by a reduction to Total Dominating Set and we provide several approximation lower bounds for subclasses of subcubic graphs. Moreover, we obtain complexity dichotomies in monogenic classes for the decision versions of Semitotal Dominating Set and Total Dominating Set.
Finally, we show that it is NP-complete to recognise the graphs such that γt2(G)=γt(G) and those such that γ(G)=γt2(G), even if restricted to be planar and with maximum degree at most 4, and we provide forbidden induced subgraph characterisations for the graphs hereditarily satisfying either of these two equalities.