On the word-representability of $K_m$-$K_n$ graphs
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.
Cite
@article{arxiv.2508.15177,
title = {On the word-representability of $K_m$-$K_n$ graphs},
author = {Herman Z. Q. Chen and Humaira Hameed and Sergey Kitaev},
journal= {arXiv preprint arXiv:2508.15177},
year = {2025}
}
Comments
To appear in the Discussiones Mathematicae Graph Theory