Bounds on the Upper \(k\)-Domination Number and the Upper \(k\)-Star-Forming Number of a Graph

A subset \( A \) of vertices of a graph \( G \) is a \( k \)-dominating set if every vertex not in \( A \) has at least \( k \) neighbors in \( A \) and a \( k \)-star-forming set if every vertex not in \( A \) forms with \( k \) vertices of \( A \) a not necessarily induced star \( K_{1, k} \). The maximum cardinalities of a minimal \( k \)-dominating set and of a minimal \( k \)-star-forming set of \( G \) are respectively denoted by \( \Gamma_k(G) \) and \( \text{SF}_k(G) \). We determine upper bounds on \( \Gamma_k(G) \) and \( \text{SF}_k(G) \) and describe the structure of the extremal graphs attaining them.