The rank-width of the square grid

Abstract

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×n square grid, denoted by G n,n , is a graph on the vertex set , where a vertex (x,y) is connected by an edge to a vertex (x′, y′) if and only if |x -x′| + |y - y′| = 1. We prove that the rank-width of G n,n is equal to n - 1, thus solving an open problem of Oum. © 2008 Springer Berlin Heidelberg.