New results on word-representable graphs

Andrew Collins; Sergey Kitaev; Vadim V. Lozin

doi:10.1016/j.dam.2014.10.024

New results on word-representable graphs

Andrew Collins, Sergey Kitaev, Vadim V. Lozin

Research output: Contribution to journal › Article › peer-review

11 Citations (Scopus)

102 Downloads (Pure)

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
Pages (from-to)	136-141
Number of pages	6
Journal	Discrete Applied Mathematics
Volume	216
Issue number	Part 1
Early online date	20 Nov 2014
DOIs	https://doi.org/10.1016/j.dam.2014.10.024
Publication status	Published - 10 Jan 2017

Keywords

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

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10.1016/j.dam.2014.10.024

Collins-Kitaev-Lozin-DAM-2014-New-results-on-word-representable-graphsAccepted author manuscript, 305 KB

Sergey Kitaev
- sergey.kitaevstrath.acuk
- Mathematics And Statistics - Professor
Person: Academic

Cite this

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title = "New results on word-representable graphs",

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).",

keywords = "semi-transitive orientation, speed of hereditary properties, hereditary property of graphs",

author = "Andrew Collins and Sergey Kitaev and Lozin, {Vadim V.}",

note = "“NOTICE: this is the author{\textquoteright}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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AU - Collins, Andrew

AU - Kitaev, Sergey

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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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N2 - 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).

AB - 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).

KW - semi-transitive orientation

KW - speed of hereditary properties

KW - hereditary property of graphs

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DO - 10.1016/j.dam.2014.10.024

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VL - 216

SP - 136

EP - 141

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

IS - Part 1

ER -

New results on word-representable graphs

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Keywords

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Sergey Kitaev

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