The Dual-Server Domination Number of Some Graph Operations

A subset $S\subseteq V(G)$ is called a dual-server dominating set if there exists a partition ${\pi }_S=\{R,B\}$ of $S$ such that every vertex in $V(G)\setminus S$ has at least one neighbour in $R$ and at least one neighbour in $B\mathrm{.}$ The minimum cardinality of a DS-dominating set of $G$ is called the dual-server domination number, denoted by ${\gamma }_{ds}(G)$. 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.