Isolate semitotal domination in graphs

Abstract

In this paper, we introduce a new domination parameter called "Isolate Semitotal Domination". A set $S$ of vertices in a graph $G$ is said to be 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 $S$. A semitotal dominating set $S$ of a graph $G$ is said to be an "Isolate Semitotal Dominating set"(ISTD-set) of $G$ if $\langle S \rangle$ has at least one isolated vertex. The isolate semitotal domination number(ISTD-number), denoted by $\gamma_{t_2}^0 (G)$, is the minimum cardinality of an isolate semitotal dominating set of $G$. In this paper, we studied some basic properties of ISTD-set. Further, we characterize all connected graphs and trees for which the ISTD-number is $n-1$ and $n-2$, where $n$ is the order of the graph. At last, we proved that the problem of finding the ISTD-number for bipartite graphs is NP-complete.