$$K_{1,2}$$ -Isolation Number of Claw-Free Cubic Graphs

Abstract

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