The forcing nonsplit domination number of a graph

Abstract

A dominating set $S$ of a graph $G$ is said to be nonsplit dominating set if the subgraph $⟨V-S⟩$ is connected. The minimum cardinality of a nonsplit dominating set is called the nonsplit domination number and is denoted by ${\gamma }_{ns}(G)$. For a minimum nonsplit dominating set $S$ of $G$, a set $T\subseteq S$ is called a forcing subset for $S$ if $S$ is the unique ${\gamma }_{ns}$-set containing $T$. A forcing subset for $S$ of minimum cardinality is a minimum forcing subset of $S$. The forcing nonsplit domination number of $S$, denoted by $f_{\gamma ns}(S)$, is the cardinality of a minimum forcing subset of $S$. The forcing nonsplit domination number of $G$, denoted by $f_{\gamma ns}(G)$ is defined by $f_{\gamma ns}(G)=min\{f_{\gamma ns}(S)\}$, where the minimum is taken over all ${\gamma }_{ns}$-sets $S$ in $G$. The forcing nonsplit domination number of certain standard graphs are determined. It is shown that, for every pair of positive integers $a$ and $b$ with $0\le a\le b$ and $b\ge 1$, there exists a connected graph $G$ such that $f_{\gamma ns}(G)=a$ and ${\gamma }_{ns}(G)=b$. It is shown that, for every integer $a\ge 0$, there exists a connected graph $G$ with $f_{\gamma }(G)=f_{\gamma ns}(G)=a$, where $f_{\gamma }(G)$ is the forcing domination number of the graph. Also, it is shown that, for every pair $a,b$ of integers with $a\ge 0$ and $b\ge 0$ there exists a connected graph $G$ such that $f_{\gamma }(G)=a$ and $f_{\gamma ns}(G)=b$.