The 2-domination and Roman domination numbers of grid graphs

We investigate the 2-domination number for grid graphs, that is the size of a smallest set D of vertices of the grid such that each vertex of the grid belongs to D or has at least two neighbours in D. We give a closed formula giving the 2-domination number of any n x m grid, hereby confirming the results found by Lu and Xu, and Shaheen et al. for n <= 4 and slightly correct the value of Shaheen et al. for n = 5. The proof relies on some dynamic programming algorithms, using transfer matrices in (min,+)-algebra. We also apply the method to solve the Roman domination problem on grid graphs.