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
|
Text (Akgun-etal-JIS2019-Solving-computational-problems-in-the-theory-of-word-representable-graphs)
Akgun_etal_JIS2019_Solving_computational_problems_in_the_theory_of_word_representable_graphs.pdf Final Published Version Download (423kB)| Preview |
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.
Creators(s): |
Akgün, Özgür, Gent, Ian, Kitaev, Sergey ![]() | Item type: | Article |
---|---|
ID code: | 67081 |
Keywords: | 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: | 01 Jan 2021 12:58 |
URI: | https://strathprints.strath.ac.uk/id/eprint/67081 |
Export data: |