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In [10], Fink and Jacobson gave a generalization of the concepts of domination and independence in graphs which extends only partially the well-known inequality chain \( \gamma(G) \leq i(G) \leq \beta(G) \leq \Gamma(G) \) between the usual parameters of domination and independence. If a \( k \)-independent set is defined as a subset of vertices inducing in \( G \) a subgraph of maximum degree less than \( k \), we introduce the property which makes a \( k \)-independent set maximal. This leads us to the notion of a \( k \)-star-forming set. The corresponding parameters \( sf_k(G) \) and \( \text{SF}_k(G) \) satisfy \( sf_k(G) \leq i_k(G) \leq \beta_k(G) \leq \text{SF}_k(G) \) where \( i_k(G) \) and \( \beta_k(G) \) are respectively the minimum and the maximum cardinality of a maximal \( k \)-independent set. We initiate the study of \( sf_k(G) \) and \( \text{SF}_k(G) \) and give some results in particular classes of graphs such as trees, chordal graphs, and \( K_{1,r} \)-free graphs.