Security number of grid-like graphs

The security number of a graph is the cardinality of a smallest vertex subset of the graph such that any attack on the subset is defendable. In this paper, we determine the security number of two-dimensional cylinders and tori. This result settles a conjecture of Brigham et al. R.C. Brigham, R.D. Dutton, S.T. Hedetniemi, Security in graphs, Discrete Appl. Math. 155 (2007) 1708-1714].