### Abstract

Original language | English |
---|---|

Pages (from-to) | 164-171 |

Number of pages | 14 |

Journal | Discrete Applied Mathematics |

Volume | 201 |

Early online date | 24 Aug 2015 |

DOIs | |

Publication status | Published - 11 Mar 2016 |

### Fingerprint

### Keywords

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

### Cite this

*Discrete Applied Mathematics*,

*201*, 164-171. https://doi.org/10.1016/j.dam.2015.07.033

}

*Discrete Applied Mathematics*, vol. 201, pp. 164-171. https://doi.org/10.1016/j.dam.2015.07.033

**Semi-transitive orientations and word-representable graphs.** / Halldórsson, Magnús M.; Kitaev, Sergey; Pyatkin, Artem.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Semi-transitive orientations and word-representable graphs

AU - Halldórsson, Magnús M.

AU - Kitaev, Sergey

AU - Pyatkin, Artem

PY - 2016/3/11

Y1 - 2016/3/11

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.

AB - 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.

KW - graphs

KW - comparability graphs

KW - circle graphs

KW - complexity

KW - word-representability

KW - orientations

UR - http://www.sciencedirect.com/science/article/pii/S0166218X15003868

U2 - 10.1016/j.dam.2015.07.033

DO - 10.1016/j.dam.2015.07.033

M3 - Article

VL - 201

SP - 164

EP - 171

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -