On matching and semitotal domination in graphs

Abstract

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, γt(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, γt2(G), is the minimum cardinality of a semitotal dominating set of G. We observe that γ(G)≤γt2(G)≤γt(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 γt2(G)≤α′(G)+1 and we characterize the graphs achieving equality in the bound.