Graphs of bounded rank-width
Approximating clique-width and branch-width
Journal of Combinatorial Theory Series B
Rank-width is less than or equal to branch-width
Journal of Graph Theory
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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.