Neighbor-isolated scattering number of graphs

Let \(G=(V,E)\) be a graph. For a vertex \(u\) of \(V(G)\), its closed neighborhood, \(N[S]\), is defined as \(N[u]=\{u\}\cup\{v|v\in V(G), v\neq u, u\) and \(v\) are adjacent in \(G \}\). A vertex subset \(S\) of \(V(G)\) is called a subversion strategy of \(G\) if each of the vertices in \(N[S]\) is deleted from \(G\). By \(G/S\) we denote the survival subgraph \(G-N[S]\). A subversion strategy \(S\) is called a cut strategy of \(G\) if \(G/S\) is disconnected, or is a clique, or is empty. In this paper, we revise the definition of neighbor-isolated scattering number, which was introduced by Aslan, as \(NIS(G)=\max\{i(G/S)-|S|\}\), where \(S\) represents a cut strategy of \(G\) such that every component of \(G/S\) is an isolated vertex or a clique, and \(i(G/S)\) represents the number of the components of \(G/S\). We discuss the relationship between this parameter and the structure of graphs. Some tight bounds and extremal graphs with given order and neighbor-isolated scattering number are determined.