# New results on word-representable graphs

Andrew Collins, Sergey Kitaev, Vadim V. Lozin

Research output: Contribution to journalArticle

5 Citations (Scopus)

### Abstract

A graph G=(V,E)G=(V,E) is word-representable if there exists a word ww over the alphabet VV such that letters xx and yy alternate in ww if and only if (x,y)∈E(x,y)∈E for each x≠yx≠y. The set of word-representable graphs generalizes several important and well-studied graph families, such as circle graphs, comparability graphs, 3-colorable graphs, graphs of vertex degree at most 3, etc. By answering an open question from Halldórsson et al. (2011), in the present paper we show that not all graphs of vertex degree at most 4 are word-representable. Combining this result with some previously known facts, we derive that the number of nn-vertex word-representable graphs is View the MathML source2n23+o(n2).
Original language English 136-141 6 Discrete Applied Mathematics 216 Part 1 20 Nov 2014 https://doi.org/10.1016/j.dam.2014.10.024 Published - 10 Jan 2017

### Fingerprint

Graph in graph theory
Vertex Degree
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Comparability Graph
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### Keywords

• semi-transitive orientation
• speed of hereditary properties
• hereditary property of graphs

### Cite this

Collins, Andrew ; Kitaev, Sergey ; Lozin, Vadim V. / New results on word-representable graphs. In: Discrete Applied Mathematics. 2017 ; Vol. 216, No. Part 1. pp. 136-141.
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abstract = "A graph G=(V,E)G=(V,E) is word-representable if there exists a word ww over the alphabet VV such that letters xx and yy alternate in ww if and only if (x,y)∈E(x,y)∈E for each x≠yx≠y. The set of word-representable graphs generalizes several important and well-studied graph families, such as circle graphs, comparability graphs, 3-colorable graphs, graphs of vertex degree at most 3, etc. By answering an open question from Halld{\'o}rsson et al. (2011), in the present paper we show that not all graphs of vertex degree at most 4 are word-representable. Combining this result with some previously known facts, we derive that the number of nn-vertex word-representable graphs is View the MathML source2n23+o(n2).",
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New results on word-representable graphs. / Collins, Andrew; Kitaev, Sergey; Lozin, Vadim V.

In: Discrete Applied Mathematics, Vol. 216, No. Part 1, 10.01.2017, p. 136-141.

Research output: Contribution to journalArticle

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N1 - “NOTICE: this is the author’s version of a work that was accepted for publication in Discrete Applied Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Discrete Applied Mathematics, [VOL 216, Part 1, (20/11/14)] DOI: 10.1016/j.dam.2014.10.024¨

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