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arXiv · 1306.1345

A Note on Graphs of Linear Rank-Width 1

Binh-Minh Bui-Xuan·Mamadou Moustapha Kanté·Vincent Limouzy
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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].

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Binh-Minh Bui-Xuan, Mamadou Moustapha Kanté, Vincent Limouzy. 2013-06-06. A Note on Graphs of Linear Rank-Width 1. https://arxiv.org/abs/1306.1345

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