### Abstract

Also, a graph is word-representable iff it is k-representable for some k, that is, if it can be represented using k copies of each letter. The minimum such k for a given graph is called graph's representation number. Our computational results in this paper not only include distribution of k-representable graphs on at most 9 vertices, but also have relevance to a known conjecture on these graphs. In particular, we find a new graph on 9 vertices with high representation number. Also, we prove that a certain graph has highest representation number among all comparability graphs on odd number of vertices.

Finally, we introduce the notion of a k-semi-transitive orientation refining the notion of a semi-transitive orientation, and show computationally that the refinement is not equivalent to the original definition, unlike the equivalence of k-representability and word-representability.

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

Article number | 19.2.5 |

Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Journal of Integer Sequences |

Volume | 22 |

Issue number | 2 |

Publication status | Published - 24 Feb 2019 |

### Fingerprint

### Keywords

- word-representable graph
- semi-transitive orientation
- representation number
- enumeration
- k-semi-transitive orientation

### Cite this

*Journal of Integer Sequences*,

*22*(2), 1-18. [19.2.5].

}

*Journal of Integer Sequences*, vol. 22, no. 2, 19.2.5, pp. 1-18.

**Solving computational problems in the theory of word-representable graphs.** / Akgün , Özgür; Gent, Ian; Kitaev, Sergey; Zantema, Hans.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Solving computational problems in the theory of word-representable graphs

AU - Akgün , Özgür

AU - Gent, Ian

AU - Kitaev, Sergey

AU - Zantema, Hans

PY - 2019/2/24

Y1 - 2019/2/24

N2 - A simple graph G = (V, E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w iff xy ∈ E. Word-representable graphs generalize several important classes of graphs. A graph is word-representable iff it admits a semi-transitive orientation. We use semi-transitive orientations to enumerate connected non-word-representable graphs up to the size of 11 vertices, which led to a correction of a published result. Obtaining the enumeration results took 3 CPU years of computation.Also, a graph is word-representable iff it is k-representable for some k, that is, if it can be represented using k copies of each letter. The minimum such k for a given graph is called graph's representation number. Our computational results in this paper not only include distribution of k-representable graphs on at most 9 vertices, but also have relevance to a known conjecture on these graphs. In particular, we find a new graph on 9 vertices with high representation number. Also, we prove that a certain graph has highest representation number among all comparability graphs on odd number of vertices.Finally, we introduce the notion of a k-semi-transitive orientation refining the notion of a semi-transitive orientation, and show computationally that the refinement is not equivalent to the original definition, unlike the equivalence of k-representability and word-representability.

AB - A simple graph G = (V, E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w iff xy ∈ E. Word-representable graphs generalize several important classes of graphs. A graph is word-representable iff it admits a semi-transitive orientation. We use semi-transitive orientations to enumerate connected non-word-representable graphs up to the size of 11 vertices, which led to a correction of a published result. Obtaining the enumeration results took 3 CPU years of computation.Also, a graph is word-representable iff it is k-representable for some k, that is, if it can be represented using k copies of each letter. The minimum such k for a given graph is called graph's representation number. Our computational results in this paper not only include distribution of k-representable graphs on at most 9 vertices, but also have relevance to a known conjecture on these graphs. In particular, we find a new graph on 9 vertices with high representation number. Also, we prove that a certain graph has highest representation number among all comparability graphs on odd number of vertices.Finally, we introduce the notion of a k-semi-transitive orientation refining the notion of a semi-transitive orientation, and show computationally that the refinement is not equivalent to the original definition, unlike the equivalence of k-representability and word-representability.

KW - word-representable graph

KW - semi-transitive orientation

KW - representation number

KW - enumeration

KW - k-semi-transitive orientation

UR - https://cs.uwaterloo.ca/journals/JIS/VOL22/Kitaev/kitaev11.html

M3 - Article

VL - 22

SP - 1

EP - 18

JO - Journal of Integer Sequences

JF - Journal of Integer Sequences

SN - 1530-7638

IS - 2

M1 - 19.2.5

ER -