Representation Number of Word-Representable Split Graphs
Tithi Dwary, Khyodeno Mozhui, K. V. Krishna
TL;DR
This paper proves that the representation number for word-representable split graphs is always at most $3$ and provides a concrete algorithm to construct a $3$-uniform word that represents any such graph, leveraging a semi-transitive orientation and a labeled clique. It then characterizes when the representation number is exactly $3$, introducing the induced-subgraph framework via the class $\mathcal{C}_3$ and detailing which graphs within this class—such as $F_0$ and $F_1(5)$—determine the case for split comparability graphs. The work links word-representability of split graphs to forbidden subgraph structures and clarifies the landscape for split comparability graphs by giving a precise $\mathcal{R}=3$ criterion. Overall, the results advance understanding of representation numbers in restricted graph families and set the stage for a complete forbidden-subgraph characterization of word-representable split graphs.
Abstract
A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. The word-representability of split graphs was studied in a series of papers in the literature, and the class of word-representable split graphs was characterized through semi-transitive orientation. Nonetheless, the representation number of this class of graphs is still not known. In general, determining the representation number of a word-representable graph is an NP-complete problem. In this work, through an algorithmic procedure, we show that the representation number of the class of word-representable split graphs is at most three. Further, we characterize the class of word-representable split graphs as well as the class of split comparability graphs which have representation number exactly three.