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There is a long line of research in the literature dedicated to wordrepresentable 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 wordrepresentable graph. The two surprisingly simple solutions provided by us answer a question that was open for about ten years.

A graph G = (V, E) is word-representable if there exists a word w over the alphabet V such that letters x and y, x = y, alternate in w if and only if (x, y) ∈ E. Halldórsson et al. have shown that a graph is wordrepresentable if and only if it admits a so-called semi-transitive orientation.A corollary to this result is that any 3-colorable graph is word-representable.Akrobotu et al. have shown that a triangulation of a grid graph is wordrepresentable if and only if it is 3-colorable. This result does not hold for triangulations of grid-covered cylinder graphs, namely, there are such wordrepresentable graphs with chromatic number 4. In this paper we show that word-representability of triangulations of grid-covered cylinder graphs with three sectors (resp., more than three sectors) is characterized by avoiding a certain set of six minimal induced subgraphs (resp., wheel graphs W 5 and W 7 ).

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