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arXiv · 2508.15177

On the word-representability of $K_m$-$K_n$ graphs

Herman Z. Q. Chen·Humaira Hameed·Sergey Kitaev
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Abstract

Word-representable graphs are a class of graphs that can be represented by words, where edges and non-edges are determined by the alternation of letters in those words. Several papers in the literature have explored the word-representability of split graphs, in which the vertices can be partitioned into a clique and an independent set. In this paper, we initiate the study of the word-representability of graphs in which the vertices can be partitioned into two cliques. We provide a complete characterization of such word-representable graphs in terms of forbidden subgraphs when one of the cliques has a size of at most four. In particular, if one of the cliques is of size four, we prove that there are seven minimal non-word-representable graphs.

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Herman Z. Q. Chen, Humaira Hameed, Sergey Kitaev. 2025-08-21. On the word-representability of $K_m$-$K_n$ graphs. https://arxiv.org/abs/2508.15177

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