Word-Representation of Melon Graphs

2026-02-15

AI Summary

This study addresses the incomplete characterization of graph classes with representation number three, particularly focusing on melon graphs and their line graphs, whose representational capabilities within the frameworks of word-representability and comparability remain unclear. We prove for the first time that melon graphs have representation number at most three, characterize their structural properties when they are comparability graphs, and fully describe the word-representability of their line graphs. By integrating graph-theoretic, combinatorial representation-theoretic, and algebraic methods—alongside techniques from comparability graph theory and permutation representation analysis—we establish that melon graphs, their comparability subclasses, and their line graphs all have representation number no greater than three. This work thus provides a new, explicitly characterizable family of graphs with representation number three.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Representation Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for 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

The notion of word-representable graphs is a generalization of comparability graphs, in which graphs are represented by words. The complexity of word-representation of a word-representable graph is captured through the representation number, whereas the corresponding concept is the permutation-representation number for comparability graphs. The graphs with the (permutation-)representation number at most two were characterized in the literature. While certain examples in the class of graphs with the (permutation-)representation number three are known, no characterization for these classes is available. In this work, we prove that the representation number of melon graphs is at most three. Further, we characterize the class of melon graphs restricted to comparability graphs and show that their permutation-representation number is also at most three. Moreover, this work characterizes the word-representable line graphs of melon graphs and establishes that their representation number is at most three.

Problem

• word-representable graphs
• representation number
• melon graphs
• comparability graphs
• line graphs

Innovation

• word-representable graphs
• representation number
• melon graphs
• comparability graphs
• line graphs