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 $\{1,2,\dotsc,n\}\times\{1,2,\dotsc,n\}$, 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.