K_1,2-Isolation Number of Claw-Free Cubic Graphs

Let G be a graph and F be a family of connected graphs. A subset S of G is called an F-isolating set if G - N[S] contains no member in F as a subgraph, and the minimum cardinality of an F-isolating set of graph G is called the F-isolation number of graph G, denoted by iota(G, F). For simplicity, let iota(G, {K-1,K- 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) <= n/4. The bound is sharp.