\(K_{1,2}\)-Isolation Number of Claw-Free Cubic Graphs

Let G be a graph and \({\mathcal {F}}\) $\mathcal{F}$ be a family of connected graphs. A subset S of G is called an \({\mathcal {F}}\) $\mathcal{F}$ -isolating set if \(G-N[S]\) $G-N[S]$ contains no member in \({\mathcal {F}}\) $\mathcal{F}$ as a subgraph, and the minimum cardinality of an \({\mathcal {F}}\) $\mathcal{F}$ -isolating set of graph G is called the \({\mathcal {F}}\) $\mathcal{F}$ -isolation number of graph G, denoted by \(\iota (G,{\mathcal {F}})\) $\iota (G,\mathcal{F})$ . For simplicity, let \(\iota (G,\{K_{1,k+1}\})=\iota _k(G)\) $\iota (G,\{K_{1,k+1}\})={\iota }_k(G)$ . Thus, \(\iota _1(G)\) ${\iota }_1(G)$ is the cardinality of a smallest set S such that \(G-N[S]\) $G-N[S]$ consists of \(K_1\) $K_1$ and \(K_2\) $K_2$ only. In this paper, we prove that for any claw-free cubic graph G of order n, \(\iota _1(G)\le \frac{n}{4}\) ${\iota }_1(G)\le \frac{n}{4}$ . The bound is sharp.