Solving computational problems in the theory of word-representable graphs [article]

Özgür Akgün, Ian P. Gent, Sergey Kitaev, Hans Zantema
2018 arXiv   pre-print
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
more » ... 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. 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.
arXiv:1808.01215v1 fatcat:yzl3vjtnajce3jxyi2g4nxwyna