Representation Number 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.
Cite
@article{arxiv.2502.00872,
title = {Representation Number of Word-Representable Split Graphs},
author = {Tithi Dwary and Khyodeno Mozhui and K. V. Krishna},
journal= {arXiv preprint arXiv:2502.00872},
year = {2025}
}
Comments
The graphs in Fig. 2 are corrected using the PhD thesis of N. Pardal [22]. Accordingly, updated the proof of Theorem 6 (the characterization of word-representable split graphs with representation number three) and subsequent results on the characterization of split comparability graphs with representation number three