Isolation in graphs

Let G be a graph and F a family of graphs. We say a subset S of the vertices of G to be F-isolating if the graph induced by the vertices outside S that have no neighbors in S contains no member of F as a subgraph. The F-isolation number ι(G,F) of a graph G is the minimum cardinality among all its F-isolating sets. In this paper, we discuss several properties of the F-isolation number and then center our attention on the family F={K2}, for which the F-isolation number is called simply isolation number, we give some sharp upper bounds on the isolation number for general graphs, connected graphs, bipartite graphs, trees and outerplanar graphs.