  We prove that every graph of rank-width $k$ is a pivot-minor of a graph of
tree-width at most $2k$. We also prove that graphs of rank-width at most 1,
equivalently distance-hereditary graphs, 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.
