Representation Number of Word-Representable Split Graphs

TL;DR

This paper investigates the representation number of word-representable split graphs, establishing an upper bound of three and characterizing those with exactly three, advancing understanding of their structural properties.

Contribution

The paper introduces an algorithmic approach to determine the representation number of word-representable split graphs and characterizes those with a number exactly three.

Findings

Representation number of split graphs is at most three.

Characterization of split graphs with representation number exactly three.

Algorithmic procedure for determining the representation number.

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.