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.