Existence of μ-representation of graphs

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Abstract

Recently, Jones et al. introduced the study of μ-representable graphs, where μ is a word over { 1,2} containing at least one 1. The notion of a μ-representable graph is a far-reaching generalization of the notion of a word-representable graph studied in the literature in a series of papers.
Jones et al. have shown that any graph is 11⋯1-representable assuming that the number of 1s is at least three, while the class of 12-rerpesentable graphs is properly contained in the class of comparability graphs, which, in turn, is properly contained in the class of word-representable graphs corresponding to 11-representable graphs. Further studies in this direction were conducted by Nabawanda, who has shown, in particular, that the class of 112-representable graphs is not included in the class of word-representable graphs.
Jones et al. raised a question on classification of μ-representable graphs at least for small values of μ. In this paper we show that if μ is of length at least 3 then any graph is μ-representable. This rather unexpected result shows that from existence of representation point of view there are only two interesting non-equivalent cases in the theory of μ-representable graphs, namely, those of μ=11 and μ=12.
Original languageEnglish
Pages (from-to)661-668
Number of pages8
JournalJournal of Graph Theory
Volume85
Issue number3
Early online date28 Sep 2016
DOIs
Publication statusPublished - 31 Jul 2017

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Keywords

  • μ-representable graph
  • pattern matching
  • pattern avoidance
  • word-representable graphs

Cite this

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title = "Existence of μ-representation of graphs",
abstract = "Recently, Jones et al. introduced the study of μ-representable graphs, where μ is a word over { 1,2} containing at least one 1. The notion of a μ-representable graph is a far-reaching generalization of the notion of a word-representable graph studied in the literature in a series of papers. Jones et al. have shown that any graph is 11⋯1-representable assuming that the number of 1s is at least three, while the class of 12-rerpesentable graphs is properly contained in the class of comparability graphs, which, in turn, is properly contained in the class of word-representable graphs corresponding to 11-representable graphs. Further studies in this direction were conducted by Nabawanda, who has shown, in particular, that the class of 112-representable graphs is not included in the class of word-representable graphs. Jones et al. raised a question on classification of μ-representable graphs at least for small values of μ. In this paper we show that if μ is of length at least 3 then any graph is μ-representable. This rather unexpected result shows that from existence of representation point of view there are only two interesting non-equivalent cases in the theory of μ-representable graphs, namely, those of μ=11 and μ=12.",
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author = "Sergey Kitaev",
note = "This is the accepted version of the following article: Kitaev, S. (2016). Existence of μ-representation of graphs. Journal of Graph Theory. which has been published in final form at http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-0118 . This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [http://olabout.wiley.com/WileyCDA/Section/id-828039.html].",
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Existence of μ-representation of graphs. / Kitaev, Sergey.

In: Journal of Graph Theory, Vol. 85, No. 3, 31.07.2017, p. 661-668.

Research output: Contribution to journalArticle

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N2 - Recently, Jones et al. introduced the study of μ-representable graphs, where μ is a word over { 1,2} containing at least one 1. The notion of a μ-representable graph is a far-reaching generalization of the notion of a word-representable graph studied in the literature in a series of papers. Jones et al. have shown that any graph is 11⋯1-representable assuming that the number of 1s is at least three, while the class of 12-rerpesentable graphs is properly contained in the class of comparability graphs, which, in turn, is properly contained in the class of word-representable graphs corresponding to 11-representable graphs. Further studies in this direction were conducted by Nabawanda, who has shown, in particular, that the class of 112-representable graphs is not included in the class of word-representable graphs. Jones et al. raised a question on classification of μ-representable graphs at least for small values of μ. In this paper we show that if μ is of length at least 3 then any graph is μ-representable. This rather unexpected result shows that from existence of representation point of view there are only two interesting non-equivalent cases in the theory of μ-representable graphs, namely, those of μ=11 and μ=12.

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