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

TL;DR

This paper investigates the word-representability of graphs partitionable into two cliques, providing a complete characterization for cases where one clique has size at most four, including identifying minimal non-word-representable graphs.

Contribution

It introduces the study of word-representability for graphs partitionable into two cliques and characterizes such graphs with forbidden subgraphs for small clique sizes.

Findings

Complete characterization for graphs with one clique of size ≤4

Seven minimal non-word-representable graphs identified when one clique is size four

Provides forbidden subgraph criteria for word-representability

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.