Graphs of Small Rank-width are Pivot-minors of Graphs of Small Tree-width

TL;DR

This paper establishes a relationship between graphs of small rank-width and those of small tree-width, showing that graphs with bounded rank-width are pivot-minors of graphs with bounded tree-width, with special cases for rank-width 1.

Contribution

It proves that graphs of rank-width k are pivot-minors of graphs with tree-width at most 2k, and characterizes graphs of rank-width 1 as vertex-minors of trees and paths.

Findings

Graphs of rank-width k are pivot-minors of graphs with tree-width ≤ 2k.

Distance-hereditary graphs (rank-width 1) are vertex-minors of trees.

Graphs of linear rank-width 1 are vertex-minors of paths.

Abstract

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.