On the treewidth of toroidal grids

Many graph parameters of grid-like graphs are studied because of their algorithmic consequences. An important concept in this context is that of treewidth. Treewidth of graphs is a graph parameter for measuring how close a graph is to a tree. In this paper, we study the treewidth of toroidal grids and show that the treewidth of the n×n toroidal grid is either 2n−2 or 2n−1. We then show that these bounds are tight in some cases. To show the lower bounds, we construct brambles of high orders.