Isolation of Cycles and Trees in Graphs

A subset D ⊆ V (G) is called an $\mathcal{F}$-isolating set of a graph G if G−N[D] contains no subgraph isomorphic to any F ∈ $\mathcal{F}$, where $\mathcal{F}$ is a family of connected graphs. The $\mathcal{F}$-isolation number of G, denoted by ι(G, $\mathcal{F}$), is the minimum cardinality of an $\mathcal{F}$-isolating set in G. In this paper, take $\mathcal{F}$ = {C₃,K_1,3,P₄} and denote ι(G, $\mathcal{F}$) simply by ${{\iota }^{\prime }}_c$,(G) which implies that ${{\iota }^{\prime }}_c$(G) is the order of a smallest set D such that G−N[D] consists of some K₁, K₂ and P₃ only. We prove that if G is a connected graph of order n and different from C₃ or C₇, then ${{\iota }^{\prime }}_c$(G)≤ $\frac{n}{4}$.