Upper bounds on isolation parameters

Let F be a set of graphs. A set D of vertices of a graph G is F-isolating if the graph obtained by removing from G all the vertices in D and their neighbors does not have a copy of a graph in F as a subgraph. The F-isolation number of G, denoted by ι(G, F), is the size of a smallest F-isolating set of G. We establish sharp upper bounds on the F-isolation number of a tree in terms of the order and the number of leaves for the case where F consists of the star K1,k for some k≥ 1. Moreover, we characterize all trees attaining the bounds.