# On Graphs Representable by Pattern-Avoiding Words

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## Abstract

In this paper we study graphs defined by pattern-avoiding words. Word-representable graphs have been studied extensively following their introduction in 2004 and are the subject of a book published by Kitaev and Lozin in 2015. Recently there has been interest in studying graphs represented by pattern-avoiding words. In particular, in 2016, Gao, Kitaev, and Zhang investigated 132-representable graphs, that is, word-representable graphs that can be represented by a word which avoids the pattern 132. They proved that all 132-representable graphs are circle graphs and provided examples and properties of 132-representable graphs. They posed several questions, some of which we answer in this paper.

One of our main results is that not all circle graphs are 132-representable, thus proving that 132-representable graphs are a proper subset of circle graphs, a question that was left open in the paper by Gao et al. We show that 123-representable graphs are also a proper subset of circle graphs, and are different from 132-representable graphs. We also study graphs represented by pattern-avoiding 2-uniform words, that is, words in which every letter appears exactly twice.

[1] A.L.L. Gao, S. Kitaev and P.B. Zhang, On 132-representable graphs, Australas. J. Combin. 69 (2017) 105–118.

[2] M.M. Halldórsson, S. Kitaev and A. Pyatkin, Semi-transitive orientations and word-representable graphs, Discrete Appl. Math. 201 (2016) 164–171. doi:10.1016/j.dam.2015.07.033

[3] S. Kitaev and A. Pyatkin, On representable graphs, J. Autom. Lang. Comb. 13 (2008) 45–54.

[4] S. Heubach and T. Mansour, Combinatorics of Compositions and Words (CRC Press, 2009). doi:10.1201/9781420072686

[5] S. Kitaev, Patterns in Permutations and Words (Springer Science & Business Media, 2011). doi:10.1007/978-3-642-17333-2

[6] S. Kitaev and V. Lozin, Words and Graphs (Springer, NY, 2015). doi:10.1007/978-3-319-25859-1

# Discussiones Mathematicae Graph Theory

## The Journal of University of Zielona Góra

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Target Group

researchers in the fields of: colourings, partitions (general colourings), hereditary properties, independence and dominating structures (sets, paths, cycles, etc.), cycles, local properties, products of graphs

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