Geodesic
Representing Split Graphs by Words
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 4, pp. 1263-1280

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There is a long line of research in the literature dedicated to word-representable graphs, which generalize several important classes of graphs. However, not much is known about word-representability of split graphs, another important class of graphs. In this paper, we show that threshold graphs, a subclass of split graphs, are word-representable. Further, we prove a number of general theorems on word-representable split graphs, and use them to characterize computationally such graphs with cliques of size 5 in terms of nine forbidden subgraphs, thus extending the known characterization for word-representable split graphs with cliques of size 4. Moreover, we use split graphs, and also provide an alternative solution, to show that gluing two word-representable graphs in any clique of size at least 2 may, or may not, result in a word-representable graph. The two surprisingly simple solutions provided by us answer a question that was open for about ten years.
Keywords: split graph, word-representability, semi-transitive orientation 
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     title = {Representing {Split} {Graphs} by {Words}},
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Chen, Herman Z.Q.; Kitaev, Sergey; Saito, Akira. Representing Split Graphs by Words. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 4, pp. 1263-1280. http://geodesic.mathdoc.fr/item/DMGT_2022_42_4_a14/