On the complexity of strong and weak total domination in graphs

A set D of vertices in a graph G = (V,E) is a total dominating set if every vertex of G is adjacent to some vertex in D. A total dominating set D of G is said to be weak if every vertex v ∈ V −D is adjacent to a vertex u ∈ D such that dG(v) ≥ dG(u). The weak total domination number γwt(G) of G is the minimum cardinality of a weak total dominating set of G. A total dominating set D of G is said to be strong if every vertex v ∈ V −D is adjacent to a vertex u ∈ D such that dG(u) ≥ dG(v). The strong total domination number γst(G) of G is the minimum cardinality of a strong dominating set of G. We show that the decision problems for these variants are NP-complete, even when restricted to bipartite graphs and chordal graphs. We also show that the decision problem for the strong total domination is NP-complete, even when restricted to split graphs.