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

Akgün, Özgür and Gent, Ian and Kitaev, Sergey and Zantema, Hans
(2019)
*Solving computational problems in the theory of word-representable graphs.*
Journal of Integer Sequences, 22 (2).
pp. 1-18.
19.2.5.
ISSN 1530-7638

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## Abstract

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.

#### ORCID iDs

Akgün, Özgür, Gent, Ian, Kitaev, Sergey ORCID: https://orcid.org/0000-0003-3324-1647 and Zantema, Hans;Item type: Article ID code: 67081 Dates: DateEvent24 February 2019Published24 February 2019AcceptedKeywords: word-representable graph, semi-transitive orientation, representation number, enumeration, k-semi-transitive orientation, Electronic computers. Computer science, Computer Science(all) Subjects: Science > Mathematics > Electronic computers. Computer science Department: Faculty of Science > Computer and Information Sciences Depositing user: Pure Administrator Date deposited: 25 Feb 2019 14:12 Last modified: 28 Mar 2021 01:47 URI: https://strathprints.strath.ac.uk/id/eprint/67081