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

Özgür Akgün, Ian Gent, Sergey Kitaev, Hans Zantema

### 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.
Original language English 19.2.5 18 Journal of Integer Sequences 22 2 Published - 2019

### Fingerprint

Graph in graph theory
Representability
Comparability Graph
Graph Representation
Odd number
Simple Graph
Alternate
Enumeration
Computational Results
Refinement
Equivalence
Generalise

### Keywords

• Enumeration
• K-semi-transitive orientation
• Representation number
• Semi-transitive orientation
• Word-representable graph

### Cite this

Akgün, Özgür ; Gent, Ian ; Kitaev, Sergey ; Zantema, Hans. / Solving computational problems in the theory of word-representable graphs. In: Journal of Integer Sequences. 2019 ; Vol. 22, No. 2.
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Solving computational problems in the theory of word-representable graphs. / Akgün, Özgür; Gent, Ian; Kitaev, Sergey; Zantema, Hans.

In: Journal of Integer Sequences, Vol. 22, No. 2, 19.2.5, 2019.

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

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