A dynamic domination problem in graphs is considered in which an infinite sequence of attacks occur at vertices with mobile guards; the guard at the attacked vertex is required to vacate the vertex by moving to a neighboring vertex with no guard. Other guards are allowed to move at the same time, and before and after each attack, the vertices containing guards must form a dominating set of the graph. The minimum number of guards that can defend the graph against such an arbitrary sequence of attacks is called the m-eviction number of the graph. In this paper, the m-eviction number is determined exactly for $m \times n$ grids with $m \leq 4$ and upper bounds are given for all $n \geq m \geq 8$.
Klostermeyer, William; Messinger, Margaret-Ellen; and Angeli Ayello, Alejandro
"An Eternal Domination Problem in Grids,"
Theory and Applications of Graphs: Vol. 4
, Article 2.
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol4/iss1/2