[formula omitted]-isolation in graphs

Abstract

Let G be a graph and H be a connected graph. A subset D⊆V(G) is called an H-isolating set of G if G−N[D] contains no H as a subgraph. The H-isolation number of G, denoted by ι(G,H), is the minimum cardinality of an H-isolating set in G. For an integer k≥0, a K1,k+1-isolating set of G is simply said to be a k-isolating set of G, and ι(G,K1,k+1) is abbreviated to ιk(G), called the k-isolation number of G. Thus, ιk(G) is the minimum cardinality of a k-isolating set D such that Δ(G−N[D])≤k. We prove that if G is a connected graph of size m, then ιk(G)≤m+1k+3, unless G≅K1,k+1, or k=1 and G≅C6. The extremal graphs are completely characterized. Moreover, it is proved that for any graph G with s components, ι(G,H)≤γ(H)m(G)+sm(H)+1. Several conjectures are proposed in the closing, where γ(H) is the domination number of H.