Further results on nonsplit dom strong domination number

Abstract

A subset $S$ of $V$ is called a dom strong dominating set if for every vertex $v \in V-S$, there exists $u_1, u_2 \in S$ such that $u_1 v, u_2 v \in E(G)$ and $d\left(u_1\right) \geq d(v)$. The minimum cardinality of a dom strong dominating set is called the dom strong domination number and is denoted by $\gamma_{d s}(G)$. A dom strong dominating set $S$ is said to be a non split dom strong dominating set if the induced subgraph $\langle V-S\rangle$ is connected. The minimum cardinality of a non split dom strong dominating set is called the non split dom strong domination number of a graph and is denoted by $\gamma_{n s d s}(G)$. The connectivity $\kappa(G)$ of a graph $G$ is the minimum number of vertices whose removal results in a disconnected or trivial graph. In this paper, we find an upper bound for the sum of nonsplit dom strong domination number and connectivity of a graph and characterise the corresponding extremal graphs.