Further results on the j-independence number of graphs

In a graph G of minimum degree δ and maximum degree ∆, a subset S of vertices of G is j-independent, for some positive integer j, if every vertex in S has at most j − 1 neighbors in S. The j-independence number βj (G) is the maximum cardinality of a j-independent set of G. We first establish an inequality between βj (G) and β∆(G) for 1 ≤ j ≤ δ−1. Then we characterize all graphs G with βj (G) = β∆(G) for j ∈ {1, . . . , ∆−1}, where the particular cases j = 1, 2, δ−1 and δ are well distinguished.