Formula for integer {2}-domination numbers of four-row grids

Establish that the integer {2}-domination number of the four-row grid graph G_{4,n}=P_4\square P_n satisfies γ^{\{2\}}(G_{4,n})=2n-\lfloor n/10\rfloor for every integer n≥49.

Background

The proposed algorithm computes γ{{2}}(G_{4,n}) for n≤100, and the resulting values exhibit a regular pattern beginning at n=49. Based on these computed values, the paper conjectures the formula 2n−⌊n/10⌋ for all n≥49. The conjecture extends the finite computational data to an unrestricted range of grid lengths and remains unproved in the paper.

References

The value of γ{2}(G4,n) seems to be regular as n increases. Hence, we observe the result in Theorem 3.6 and give a conjecture. Conjecture 3.7. If n ≥ 49, thenγ{2}(G4,n) = 2n −⌊n −10⌋.

— The integer $\{2\}$-domination number of grids  (2502.00134 - Lee et al., 31 Jan 2025) in Conjecture 3.7, Section 3, p. 17