Semitotal domination versus domination and total domination in trees

Abstract

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, γt2(G), is the minimum cardinality of a semitotal dominating set of G. Clearly, γ(G) ≤ γt2(G) ≤ γt(G). In this paper, for any nontrivial tree T that is not a star, we investigate the ratios γt2(T )/γ(T) and γt(T )/γt2(T), and provide constructive characterizations of trees achieving the upper bounds.