A Note on Graphs of Linear Rank-Width 1

TL;DR

This paper characterizes graphs with linear rank-width 1 as distance-hereditary graphs with a path split decomposition tree, providing linear-time decision algorithms and several equivalent characterizations.

Contribution

It establishes a complete characterization of linear rank-width 1 graphs, including a linear-time recognition algorithm and multiple equivalent conditions.

Findings

Graphs with linear rank-width 1 are exactly the distance-hereditary graphs with a path split decomposition.

Recognition of such graphs can be done in linear time.

Several characterizations of these graphs are provided, including local equivalence to caterpillars and forbidden vertex-minors.

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].