Vertices Contained In All Or In No Minimum Semitotal Dominating Set Of A Tree

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 a graph 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_t2(G), is the minimum cardinality of a semitotal dominating set of G. We observe that γ(G) ≤ γ_t2(G) ≤ γ_t(G). We characterize the set of vertices that are contained in all, or in no minimum semitotal dominating set of a tree.