AI 中文总结
本文研究若干图运算的双服务器控制数，确定了联、笛卡尔积、 corona 图及路、圈、完全二部图分裂图的对应控制数，拓展了双服务器控制的研究。
AI 中文摘要
子集$S \subseteq V(G)$称为双服务器控制集，若存在$S$的划分$\pi_S=\{R,B\}$，使得$V(G)\setminus S$中的每个顶点都至少有一个邻居在$R$中，且至少有一个邻居在$B$中。图$G$的双服务器控制集的最小基数称为双服务器控制数，记为$\gamma_{ds}(G)$。本文研究若干图运算的双服务器控制数，确定了图的联、笛卡尔积、 corona 图的双服务器控制数的精确值，还确定了路、圈、完全二部图的分裂图的双服务器控制数，这些结果拓展了双服务器控制的研究，为其在图运算下的性质提供了进一步见解。
英文摘要
A subset $S \subseteq V(G)$ is called a dual-server dominating set if there exists a partition $π_S=\{R,B\}$ of $S$ such that every vertex in $V(G)\setminus S$ has at least one neighbour in $R$ and at least one neighbour in $B.$ The minimum cardinality of a DS-dominating set of $G$ is called the dual-server domination number, denoted by $γ_{ds}(G)$. In this paper, we investigate the dual-server domination number of several graph operations. We establish exact values for the dual-server domination number of the join, Cartesian product, and corona of graphs. We also determine the dual-server domination number of the splitting graphs of paths, cycles, and complete bipartite graphs. These results extend the study of dual-server domination and provide further insight into its behaviour under graph operations.
