The exact $(k+1)$-metric dimension of three-dimensional grids

Let $k,n_1,n_2,n_3\ge 2$, and let $P_{n_1}\Box P_{n_2}\Box P_{n_3}$ be the three-dimensional grid graph. Write $\alpha_m(n_1,n_2,n_3)$ and $\alpha_M(n_1,n_2,n_3)$ for the quantities defined in the paper, and let $\dim_{k+1}$ denote the $(k+1)$-metric dimension.

The conjecture. If

$$\alpha_m(n_1,n_2,n_3)\le k<\alpha_M(n_1,n_2,n_3),$$

then

$$\dim_{k+1}(P_{n_1}\Box P_{n_2}\Box P_{n_3})=\min\left\{4k-2\alpha_m(n_1,n_2,n_3)+4,\ n_1n_2n_3-(n_1-2)(n_2-2)(n_3-2)\right\}.$$

The preceding results determine the exact value when $k<\alpha_m(n_1,n_2,n_3)$ and show that the displayed upper bound holds throughout the remaining range where a $(k+1)$-resolving set exists. The conjecture proposes the exact value in the intermediate range $\alpha_m(n_1,n_2,n_3)\le k<\alpha_M(n_1,n_2,n_3)$; when $\alpha_m=\alpha_M$, which occurs if $\min\{n_1,n_2,n_3\}=2$, the metric dimension is already completely determined.