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 maximumcardinality 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.