发表机构
Faculty of Electronics, Telecommunications and Informatics; Gdańsk University of Technology（电子、电信与信息学院; 格但斯克理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究正则图中的彩虹 $k$-支配问题，建立上下界并证明其 NP-完全性，给出循环图等类的精确公式，并为广义彼得森图设计动态规划算法。
AI 中文摘要
彩虹 $k$-支配函数为简单图的每个顶点分配空集或颜色 $\{1\},\{2\},\ldots,\{k\}$ 中的恰好一种颜色，使得每个接收 $\emptyset$ 的顶点在其邻域中看到所有 $k$ 种颜色。该模型具有多种设施选址解释，其中彩色顶点代表提供不同类型服务的站点，而未着色顶点代表必须访问所有这些服务的受益者。在本文中，我们研究正则图中的这种支配变体。首先，我们为此类图建立上下界，并探讨其结构后果。然后，我们利用这些结果证明该问题决策形式的 NP-完全性。此外，我们推导出循环图、莫比乌斯阶梯和棱柱图的彩虹 $k$-支配数的精确公式。最后，我们为形式为 $P(6t, t)$ 的广义彼得森图开发了一种动态规划算法，并根据其实现输出及相关的整数线性规划计算提出两个猜想。
英文摘要
A rainbow $k$-dominating function assigns the empty set or exactly one of the colors $\{1\},\{2\},\ldots,\{k\}$ to each vertex of a simple graph in such a way that every vertex receiving $\emptyset$ sees all $k$ colors in its neighborhood. This model has several facility-location interpretations where colored vertices represent sites hosting different types of services, and uncolored vertices represent the beneficiaries of those services who must have access to all of them. In this paper we study this domination variant in regular graphs. First, we develop lower and upper bounds for this class of graphs, and explore their structural consequences. Then, we use those results to prove NP-completeness of a decision formulation of this problem. Additionally, we derive exact formulas for the rainbow $k$-domination number of cycles, Möbius ladders and prisms. Lastly, we develop a dynamic-programming algorithm for generalized Petersen graphs of the form $P(6t, t)$ and state two conjectures based on the output of its implementation and related integer linear programming computations.
