Semitotal Domination Multisubdivision Number of a Graph

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, $$\gamma _{t2}(G)$$ , is the minimum cardinality of a semitotal dominating set of G. The semitotal domination multisubdivision number of a graph G, $$msd_{\gamma _{t2}}(G)$$ , is the minimum positive integer k such that there exists an edge which must be subdivided k times to increase the semitotal domination number of G. In this paper, we show that $$msd_{\gamma _{t2}}(G)\le 3$$ for any graph G of order at least 3, we also determine the semitotal domination multisubdivision number for some classes of graphs and characterize trees T with $$msd_{\gamma _{t2}}(T)=1$$ , 2 and 3, respectively.