Hop Independent Sets in Graphs

Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. A set S subset of V (G) is a hop independent set of G if any two distinct vertices in S are not at a distance two from each other, that is, d(G) (v, omega) not equal 2 for any distinct vertices v, w is an element of S. The maximum cardinality of a hop independent set of G, denoted by alpha(h) (G), is called the hop independence number of G. In this paper, we show that the absolute difference of the independence number and the hop independence number of a graph can be made arbitrarily large. Furthermore, we determine the hop independence numbers of some graphs including those resulting from some binary operations of graphs.