Representation Number of Word-Representable Split Graphs

2025-02-02

AI Summary

This paper investigates the word-representability number—the minimum $k$ such that a graph is $k$-word-representable—of split graphs and their subclass, split comparability graphs. Employing a novel integration of semi-transitive orientation theory, clique–independent set structural decomposition, and combinatorial construction algorithms, we establish, for the first time, that every word-representable split graph has representability number at most 3, thereby resolving the long-standing open problem concerning the tight upper bound on this parameter for split graphs. Furthermore, we provide a complete structural characterization of those split graphs whose representability number is exactly 3, giving necessary and sufficient combinatorial conditions. Our results not only yield a tight upper bound on the word-representability complexity of split graphs but also introduce a new methodological framework and essential technical tools for the intersection of comparability graph theory and word-representability theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Applications

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• split graphs
• word representations
• comparability graphs

Innovation

• split graphs
• three-word representation
• comparability graphs