Connecting rank-width and tree-width via pivot-minors

Abstract

We prove that every graph of rank-width $k$ is a pivot-minor of a graph of tree-width at most $2k$ and every graph of linear rank-width $k$ is a pivot-minor of a graph of path-width at most $k+1$. We also prove that graphs of rank-width at most $1$ are exactly vertex-minors of trees, and graphs of linear rank-width at most $1$ are precisely vertex-minors of paths. In addition, we show that bipartite graphs of rank-width at most $1$ are exactly pivot-minors of trees and bipartite graphs of linear rank-width at most $1$ are precisely pivot-minors of paths.

Also, we calculate the linear rank-width of complete binary trees. And we show that if a tree has linear rank-width at least $k$, then it has the complete binary tree of height $k$ as a vertex-minor. Finally, we describe an interesting question that a class of graphs with unbounded linear rank-width contains all trees as vertex-minors or pivot-minors. We prove that this question for pivot-minor is false and we suggest another version of the question.