On matching and semitotal domination in graphs

Let G be a graph with no isolated vertex. In this paper, we study a parameter that is squeezed between arguably the two most important domination parameters, namely the domination number, γ ( G ) , and the total domination number, γ t ( G ) . A set S of vertices in G is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S . The semitotal domination number, γ t 2 ( G ) , is the minimum cardinality of a semitotal dominating set of G . We observe that γ ( G ) ≤ γ t 2 ( G ) ≤ γ t ( G ) . It is known that γ ( G ) ≤ α ′ ( G ) , where α ′ ( G ) denotes the matching number of G . However, the total domination number and the matching number of a graph are generally incomparable. We provide a characterization of minimal semitotal dominating sets in graphs. Using this characterization, we prove that if G is a connected graph on at least two vertices, then γ t 2 ( G ) ≤ α ′ ( G ) + 1 and we characterize the graphs achieving equality in the bound.