AI 中文总结
本文研究图的高效k受限广播支配参数，针对树开发了确定最小代价的多项式时间动态规划算法，证明任意图的k-ELDB存在性判定为NP完全问题，明确了该问题的复杂度边界。
AI 中文摘要
高效k受限支配广播（即k-ELDB）是一种k受限广播，其中每个顶点恰好被支配一次。该概念将高效支配与受限广播支配统一到同一框架中。对于图G，我们用mcr(G)表示G存在k-ELDB的最小整数k；对于容许值k≥mcr(G)，记γ_{ebk}(G)为G上k-ELDB的最小代价，称为G的k-高效广播支配数。本文从算法视角结合复杂度分析研究这些参数，针对树开发了一种动态规划算法，该算法对固定k可计算γ_{ebk}(T)，从而得到确定mcr(T)的多项式时间方法；相反，证明对于每个固定整数k≥1，判定任意图是否存在k-ELDB是NP完全问题。这些结果将高效受限广播支配置于自然的复杂度框架中，其中树构成易处理类，而任意图仍具计算难度。
英文摘要
An efficient $k$-limited dominating broadcast, or $k$-ELDB, is a $k$-limited broadcast in which every vertex is dominated exactly once. This notion brings together efficient domination and limited broadcast domination in a common framework. For a graph $G$, we write $mcr(G)$ for the smallest integer $k$ for which $G$ admits a $k$-ELDB. For an admissible value $k\ge mcr(G)$, we denote by $γ_{ebk}(G)$ the minimum cost of a $k$-ELDB on $G$, and is called the $k$-efficient broadcast domination number of $G$. In this paper, we study these parameters from an algorithmic perspective with complexity analysis. We develop a dynamic programming algorithm for trees which, for fixed $k$, computes $γ_{ebk}(T)$ and thereby obtains a polynomial-time procedure for determining $mcr(T) $. In contrast, we prove that, for every fixed integer $k\ge 1$, deciding whether a graph admits a $k$-ELDB is NP-complete for arbitrary graphs. These results place efficient limited broadcast domination in a natural complexity framework, with trees forming a tractable class and arbitrary graphs remaining computationally hard.
Comments10 pages, 2 figures
