On the safe set of Cartesian product of two complete graphs

For a connected graph $G$, a vertex subset $S$ of $V(G)$ is a safe set if for every component $C$ of the subgraph of $G$ induced by $S$, $|C| \ge |D|$ holds for every component $D$ of $G-S$ such that there exists an edge between $C$ and $D$, and, in particular, if the subgraph induced by $S$ is connected, then $S$ is called a connected safe set. For a connected graph $G$, the safe number and the connected safe number of $G$ are the minimum among sizes of the safe sets and the minimum among sizes of the connected safe sets, respectively, of $G$. Fujita et al. introduced these notions in connection with a variation of the facility location problem. In this paper, we study the safe number and the connected safe number of Cartesian product of two complete graphs. Figuring out a way to reduce the number of components to two without changing the size of safe set makes it sufficient to consider only partitions of an integer into two parts without which it would be much more complicated to take care of all the partitions. In this way, we could show that the safe number and the connected safe number of Cartesian product of two complete graphs are equal and present a polynomial-time algorithm to compute them. Especially, in the case where one of complete components has order at most four, we precisely formulate those numbers.

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