On kk-domination and jj-independence in graphs

Let GG be a graph and let kk and jj be positive integers. A subset DD of the vertex set of GG is a kk-dominating set   if every vertex not in DD has at least kk neighbors in DD. The kk-domination number  γk(G)γk(G) is the cardinality of a smallest kk-dominating set of GG. A subset I⊆V(G)I⊆V(G) is a jj-independent set   of GG if every vertex in II has at most j−1j−1 neighbors in II. The jj-independence number  αj(G)αj(G) is the cardinality of a largest jj-independent set of GG. In this work, we study the interaction between γk(G)γk(G) and αj(G)αj(G) in a graph GG. Hereby, we generalize some known inequalities concerning these parameters and put into relation different known and new bounds on kk-domination and jj-independence. Finally, we will discuss several consequences that follow from the given relations, while always highlighting the symmetry that exists between these two graph invariants.