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 , … , n } × { 1 , 2 , … , 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.