Admissible Property of Graphs in Terms of Independence Number

Abstract

For a positive integer k, a graph G is said to have the property ${\mathcal {I}}_{k}$ if each component of G has independence number at most k. The ${\mathcal {I}}_{1}$-admission number of a graph G, denoted by $\eta (G,\,{\mathcal {I}}_{1})$, is the cardinality of a smallest vertex subset D such that $V(G)=N_{G}[D]$ or each component of $G-N_{G}[D]$ is a clique. In this paper, we show that for a connected graph G of order n, if $G\not \in \{P_{3},\,C_{6}\}$, then $\eta (G,\,{\mathcal {I}}_{1})\le \frac{2n}{7}$, and the bound is sharp.