Word-representability of split graphs

Kitaev, Sergey and Long, Yangjing and Ma, Jun and Wu, Hehui (2022) Word-representability of split graphs. Journal of Combinatorics, 12 (4). 725–746. ISSN 2156-3527 (https://doi.org/10.4310/JOC.2021.v12.n4.a6)

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Two letters x and y alternate in a word w if after deleting in w all letters but the copies of x and y we either obtain a word xyxy⋯ (of even or odd length) or a word yxyx⋯ (of even or odd length). A 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 if and only if xy∈E. It is known that a graph is word-representable if and only if it admits a certain orientation called semi-transitive orientation. Word-representable graphs generalize several important classes of graphs such as 3 -colorable graphs, circle graphs, and comparability graphs. There is a long line of research in the literature dedicated to word-representable graphs. However, almost nothing is known on word-representability of split graphs, that is, graphs in which the vertices can be partitioned into a clique and an independent set. In this paper, we shed a light to this direction. In particular, we characterize in terms of forbidden subgraphs word-representable split graphs in which vertices in the independent set are of degree at most 2, or the size of the clique is 4. Moreover, we give necessary and sufficient conditions for an orientation of a split graph to be semi-transitive.