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

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

2012-03-16

arxiv.org/abs/1203.3606v1

Mathematics

Combinatorics

14 pages, 7 figures

Scientific paper

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.

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