Eternal domination on 3 x n grid graphs

In the eternal dominating set problem, guards form a dominating set on a graph and at each step, a vertex is attacked. After each attack, if the guards can "move" to form a dominating set that contains the attacked vertex, then the guards have successfully defended against the attack. We wish to determine the minimum number of guards required to successfully defend against any possible sequence of attacks, the eternal domination number. Since the domination number for grid graphs has been recently determined [Goncalves et al., SIAM J. Discrete Math. 25 (2011), 1443-1453] grid graphs are a natural class of graphs to consider for the eternal dominating set problem. Though the eternal domination number has been determined for 2 x n grids and 4 x n grids, it has remained only bounded for the 3 x n grid. The results in this paper provide major improvements to both the upper and lower bounds of the eternal domination number for 3 x n grid graphs. In particular, we show the conjectured value in [Goldwasser et al., Util. Math. 91 (2013), 47-64] is too small for certain values of n.