Solving computational problems in the theory of word-representable graphs

Özgür Akgün, Ian Gent, Sergey Kitaev, Hans Zantema

Research output: Contribution to journalArticleAcademicpeer-review

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 languageEnglish
Article number19.2.5
Number of pages18
JournalJournal of Integer Sequences
Volume22
Issue number2
Publication statusPublished - 2019

Keywords

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

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