An improved lower and upper bound of the k-limited domination number

We continue the study of $k$-limited domination in graphs, a domination variant in which each vertex of a dominating set may dominate at most $k$ vertices outside the set. This concept extends classical domination by incorporating capacity constraints on dominating vertices. We improve the general lower bound for the $k$-limited domination number ${\gamma }_k^L(G)$ by employing the concept of $k$-capacitated domination. As a consequence, we characterize graphs satisfying ${\gamma }_k^L(G)=\lceil \frac{n}{k+1}\rceil$. We also refine the known upper bound for graphs with $k<\delta (G)$ using the $k$-limited packing number. Under this condition, we describe graphs attaining ${\gamma }_k^L(G)=n-k$. These results extend and unify previous investigations for the case $k=1$ and provide a complete characterization of graphs attaining the extreme values of the $k$-limited domination number under the considered assumptions.