Admissible property of graphs in terms of independence number (Q2082373)

For a graph \(G\) and a property \(\mathcal{P}\), a subset \(D\subseteq V(G)\) is said to be a \(\mathcal{P}\)-admissible set of \(G\) if \(G-N_{G}[D]\) admits the property \(\mathcal{P}\), where \(N_{G}[D]= D\cup N_{G}(D)\). The \(\mathcal{P}\)-admission number of \(G\) is the cardinality of a minimum \(\mathcal{P}\)-admissible set in \(G\). A graph \(G\) is said to have the property \(\mathcal{I}_{k}\) if the independence number of each component of \(G\) is at most \(k\). As a generalization of the domination number and isolation number, the \(\mathcal{P}\)-admission number was proposed and investigated by \textit{H. Yu} and \textit{B. Wu} [Graphs Comb. 38, No. 1, Paper No. 6, 9 p. (2022; Zbl 1479.05290)]. In this paper, the authors establish a sharp upper bound for \(\mathcal{I}_{1}\)-admission number of generally connected graphs.