In a graph $G = (V,E)$, a k-ruling set $S$ is one in which all vertices $V$ \ $S$ are at most $k$ distance from $S$. Finding a minimum k-ruling set is intrinsically linked to the minimum dominating set problem and maximal independent set problem, which have been extensively studied in graph theory. This paper presents the first known algorithm for solving all k-ruling set problems in conjunction with known minimum dominating set algorithms at only additional polynomial time cost compared to a minimum dominating set. The algorithm further succeeds for $(\alpha, \alpha - 1)$ ruling sets in which $\alpha > 1$, for which constraints exist on the proximity of vertices v $\in S$. This secondary application instead works in conjunction with maximal independent set algorithms.

翻译：在图$G = (V,E)$中，k-支配集$S$满足所有顶点$V$ \ $S$到$S$的距离最多为$k$。寻找最小k-支配集问题与最小支配集问题及极大独立集问题有着本质联系，这些在图论中已得到广泛研究。本文提出了首个已知算法，该算法能够在仅增加多项式时间成本的前提下，结合现有最小支配集算法求解所有k-支配集问题。该算法进一步适用于$(\alpha, \alpha - 1)$支配集（其中$\alpha > 1$），此类支配集对顶点v $\in S$的邻近性存在约束。该扩展应用可与极大独立集算法协同工作。