发表机构
University of Maribor, FERI; IMFM（马里博尔大学，FERI; 斯洛文尼亚数学与物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过引入k-容量支配和k-限制打包数，改进了k-限制支配数的下界与上界，并完整刻画了达到极值的图，统一了k=1时的先前结果。
AI 中文摘要
我们继续研究图中的$k$-限制支配，这是一种支配变体，其中支配集中的每个顶点至多支配集合外的$k$个顶点。该概念通过引入支配顶点的容量约束来扩展经典支配。我们利用$k$-容量支配的概念，改进了$k$-限制支配数$\gamma_k^L(G)$的一般下界。作为推论，我们刻画了满足$\gamma_k^L(G)=\lceil \frac{n}{k+1} \rceil$的图。我们还利用$k$-限制打包数，细化了当$k<\delta(G)$时图的已知上界。在此条件下，我们描述了达到$\gamma_k^L(G)=n-k$的图。这些结果扩展并统一了先前对$k=1$情形的研究，并在所考虑的假设下，提供了达到$k$-限制支配数极值的图的完整刻画。
英文摘要
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 $γ_k^L(G)$ by employing the concept of $k$-capacitated domination. As a consequence, we characterize graphs satisfying $γ_k^L(G)=\lceil \frac{n}{k+1} \rceil$. We also refine the known upper bound for graphs with $k<δ(G)$ using the $k$-limited packing number. Under this condition, we describe graphs attaining $γ_k^L(G)=n-k$. 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.
