# Semi-transitive orientations and word-representable graphs

Magnús M. Halldórsson, Sergey Kitaev, Artem Pyatkin

Research output: Contribution to journalArticle

10 Citations (Scopus)

### Abstract

A graph G=(V,E) is a \emph{word-representable graph} if there exists a word W over the alphabet V such that letters x and y alternate in W if and only if (x,y)∈E for each x≠y. In this paper we give an effective characterization of word-representable graphs in terms of orientations. Namely, we show that a graph is word-representable if and only if it admits a \emph{semi-transitive orientation} defined in the paper. This allows us to prove a number of results about word-representable graphs, in particular showing that the recognition problem is in NP, and that word-representable graphs include all 3-colorable graphs. We also explore bounds on the size of the word representing the graph. The representation number of G is the minimum k such that G is a representable by a word, where each letter occurs k times; such a k exists for any word-representable graph. We show that the representation number of a word-representable graph on n vertices is at most 2n, while there exist graphs for which it is n/2.
Original language English 164-171 14 Discrete Applied Mathematics 201 24 Aug 2015 https://doi.org/10.1016/j.dam.2015.07.033 Published - 11 Mar 2016

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Graph in graph theory
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### Keywords

• graphs
• comparability graphs
• circle graphs
• complexity
• word-representability
• orientations

### Cite this

Halldórsson, Magnús M. ; Kitaev, Sergey ; Pyatkin, Artem. / Semi-transitive orientations and word-representable graphs. In: Discrete Applied Mathematics. 2016 ; Vol. 201. pp. 164-171.
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title = "Semi-transitive orientations and word-representable graphs",
abstract = "A graph G=(V,E) is a \emph{word-representable graph} if there exists a word W over the alphabet V such that letters x and y alternate in W if and only if (x,y)∈E for each x≠y. In this paper we give an effective characterization of word-representable graphs in terms of orientations. Namely, we show that a graph is word-representable if and only if it admits a \emph{semi-transitive orientation} defined in the paper. This allows us to prove a number of results about word-representable graphs, in particular showing that the recognition problem is in NP, and that word-representable graphs include all 3-colorable graphs. We also explore bounds on the size of the word representing the graph. The representation number of G is the minimum k such that G is a representable by a word, where each letter occurs k times; such a k exists for any word-representable graph. We show that the representation number of a word-representable graph on n vertices is at most 2n, while there exist graphs for which it is n/2.",
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Semi-transitive orientations and word-representable graphs. / Halldórsson, Magnús M.; Kitaev, Sergey; Pyatkin, Artem.

In: Discrete Applied Mathematics, Vol. 201, 11.03.2016, p. 164-171.

Research output: Contribution to journalArticle

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AU - Kitaev, Sergey

AU - Pyatkin, Artem

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N2 - A graph G=(V,E) is a \emph{word-representable graph} if there exists a word W over the alphabet V such that letters x and y alternate in W if and only if (x,y)∈E for each x≠y. In this paper we give an effective characterization of word-representable graphs in terms of orientations. Namely, we show that a graph is word-representable if and only if it admits a \emph{semi-transitive orientation} defined in the paper. This allows us to prove a number of results about word-representable graphs, in particular showing that the recognition problem is in NP, and that word-representable graphs include all 3-colorable graphs. We also explore bounds on the size of the word representing the graph. The representation number of G is the minimum k such that G is a representable by a word, where each letter occurs k times; such a k exists for any word-representable graph. We show that the representation number of a word-representable graph on n vertices is at most 2n, while there exist graphs for which it is n/2.

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