Geodesic
The 2-domination and Roman domination numbers of grid graphs
Discrete mathematics & theoretical computer science, ICGT 2018, Tome 21 (2019) no. 1

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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 $n \!\times\! m$ grid, hereby confirming the results found by Lu and Xu, and Shaheen et al. for $n \leq 4$ 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. 
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     title = {The 2-domination and {Roman} domination numbers of grid graphs},
     journal = {Discrete mathematics & theoretical computer science},
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Rao, Michaël; Talon, Alexandre. The 2-domination and Roman domination numbers of grid graphs. Discrete mathematics & theoretical computer science, ICGT 2018, Tome 21 (2019) no. 1. doi: 10.23638/DMTCS-21-1-9

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