Semitotal domination of Harary graphs

Abstract

Let $G$ be a simple finite undirected and an isolate-free graph. 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 such a set. In this paper, we study the semitotal domination number for Harary graphs, which was first introduced by Frank Harary. Since Harary graphs have the maximum possible connectivity with the minimum number of edges, many researchers are interested in studying its stability properties.