Line Graphs of Non-Word-Representable Graphs are Not Always Non-Word-Representable

2025-09-03

AI Summary

This paper addresses the closure of word-representability under the line graph operation: specifically, whether the line graph of a non-word-representable graph must necessarily be non-word-representable. By constructing and analyzing Mycielski graphs of odd cycles of length at least five, the authors establish the first explicit counterexample: such Mycielski graphs are non-word-representable, yet their line graphs are word-representable. This refutes the prior conjecture and demonstrates that word-representability is not preserved under the line graph operation. The methodology integrates combinatorial graph theory with alternating sequence analysis, yielding a systematic framework for graph construction and verification. The main contributions are: (1) the first explicit construction of a non-word-representable graph whose line graph is word-representable; (2) the revelation that the line graph operation affects word-representability in a nontrivial manner; and (3) a novel perspective and a critical counterexample advancing structural characterization and closure studies of word-representable graphs.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Web data visualization

Abstract

A graph is said to be word-representable if there exists a word over its vertex set such that any two vertices are adjacent if and only if they alternate in the word. If no such word exists, the graph is non-word-representable. In the literature, there are examples of non-word-representable graphs whose line graphs are non-word-representable. However, it is an open problem to determine whether the line graph of a non-word-representable graph is always non-word-representable or not? In this work, we address the open problem by considering a class of non-word-representable graphs, viz., Mycielski graphs of odd cycles of length at least five, and show that their line graphs are word-representable.

Problem

• Determine if line graphs of non-word-representable graphs are always non-word-representable
• Investigate word-representability of line graphs derived from Mycielski graphs
• Establish whether non-word-representable graphs always produce non-word-representable line graphs

Innovation

• Line graphs of Mycielski graphs are word-representable
• Proves non-word-representable graphs can have representable line graphs
• Solves open problem using odd cycle Mycielski construction