A Note on Graphs of Linear Rank-Width 1
Abstract
We prove that a connected graph has linear rank-width 1 if and only if it is a distance-hereditary graph and its split decomposition tree is a path. An immediate consequence is that one can decide in linear time whether a graph has linear rank-width at most 1, and give an obstruction if not. Other immediate consequences are several characterisations of graphs of linear rank-width 1. In particular a connected graph has linear rank-width 1 if and only if it is locally equivalent to a caterpillar if and only if it is a vertex-minor of a path [O-joung Kwon and Sang-il Oum, Graphs of small rank-width are pivot-minors of graphs of small tree-width, arxiv:1203.3606] if and only if it does not contain the co-K_2 graph, the Net graph and the 5-cycle graph as vertex-minors [Isolde Adler, Arthur M. Farley and Andrzej Proskurowski, Obstructions for linear rank-width at most 1, arxiv:1106.2533].
Cite
@article{arxiv.1306.1345,
title = {A Note on Graphs of Linear Rank-Width 1},
author = {Binh-Minh Bui-Xuan and Mamadou Moustapha Kanté and Vincent Limouzy},
journal= {arXiv preprint arXiv:1306.1345},
year = {2014}
}
Comments
9 pages, 2 figures. Not to be published