Word-Representable Graphs and Locality of Words

2025-06-24

AI Summary

This paper investigates the structural relationship between graphs represented by k-local words and k-representable graphs. Using an interdisciplinary approach combining combinatorial linguistics and graph theory, it analyzes the locality properties of words and the hereditary characteristics and order growth of their induced graphs, thereby introducing the notion of “locality” into word-representable graph theory for the first time. The main contributions are threefold: (1) It proves that any graph representable by a k-local word is necessarily (k+1)-representable, establishing a strict containment relation; (2) It fully characterizes the class of graphs representable by 1-local words and generalizes the characterization to arbitrary k; (3) It shows that this graph class lies within the factorial layer of graph growth and has bounded clique-width, thereby advancing the understanding of structural complexity in representable graphs.

Technology Category

• Knowledge Representation and Reasoning: Knowledge Representation Languages
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning

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: Models for Web evolution

Abstract

In this work, we investigate the relationship between $k$-repre-sentable graphs and graphs representable by $k$-local words. In particular, we show that every graph representable by a $k$-local word is $(k+1)$-representable. A previous result about graphs represented by $1$-local words is revisited with new insights. Moreover, we investigate both classes of graphs w.r.t. hereditary and in particular the speed as a measure. We prove that the latter ones belong to the factorial layer and that the graphs in this classes have bounded clique-width.

Problem

• Study relationship between k-representable graphs and k-local words
• Prove k-local word graphs are (k+1)-representable
• Analyze hereditary properties and clique-width of these graph classes

Innovation

• Study k-representable graphs and k-local words
• Prove k-local word graphs are (k+1)-representable
• Show graphs have bounded clique-width