The Rank-Width of the Square Grid

Rank-width is a graph width parameter introduced by Oum and Seymour. It is known that a class of graphs has bounded rank-width if and only if it has bounded clique-width, and that the rank-width of G is less than or equal to its branch-width. The n x n square grid, denoted by G(n,n) is a graph on the vertex set {1, 2,..., n} x {1, 2,..., n}, where a vertex (x, y) is connected by an edge to a vertex (x', y') if and only if vertical bar x - x'vertical bar + vertical bar y - y'vertical bar = 1. We prove that the rank-width of G(n,n) is equal to n - 1, thus solving an open problem of Oum.