Nonseparating Independent Sets of Cartesian Product Graphs

A set of vertices $S$ of a connected graph $G$ is a nonseparating independent set if $S$ is independent and $G-S$ is connected. The nsis number $\mathcal{Z}(G)$ is the maximum cardinality of a nonseparating independent set of $G$. It is well known that computing the nsis number of graphs is NP-hard even when restricted to $4$-regular graphs. In this paper, we first present a new sufficient and necessary condition to describe the nsis number. Then, we completely solve the problem of counting the nsis number of hypercubes $Q_{n}$ and Cartesian product of two cycles $C_{m} \square C_{n}$, respectively. We show that $\mathcal{Z}(Q_{n}) = 2^{n-2}$ for $n \geq 2$, and $\mathcal{Z}(C_{m} \square C_{n}) = n + \lfloor (n+2)/4 \rfloor$ if $m = 4$, $m + \lfloor (m+2)/4 \rfloor$ if $n = 4$ and $\lfloor mn/3 \rfloor$ otherwise. Moreover, we find a maximum nonseparating independent set of $Q_{n}$ and $C_{m} \square C_{n}$, respectively.