Bounding Width on Graph Classes of Constant Diameter

2025-05-26

AI Summary

This work investigates how diameter constraints affect the boundedness of width parameters—specifically treedepth, pathwidth, treewidth, and cliquewidth—in graph classes excluding a fixed forbidden subgraph (F) and having diameter at most (d). Method: Integrating structural graph theory, extremal graph theory, and parameterized complexity analysis, the study develops the first general criterion for when diameter restrictions transform otherwise unbounded width parameters into bounded ones. Contribution/Results: It provides a complete characterization of treedepth boundedness for (F)-free graphs of diameter (leq d) for all (d geq 4), partial classifications for (d = 2,3), and a decidable criterion for treedepth boundedness in diameter-constrained (F)-free graphs. The results unify geometric (diameter) and structural (forbidden subgraph) constraints within width parameter theory, establishing a foundational framework for algorithm design and complexity analysis on restricted graph classes.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We determine if the width of a graph class ${cal G}$ changes from unbounded to bounded if we consider only those graphs from ${cal G}$ whose diameter is bounded. As parameters we consider treedepth, pathwidth, treewidth and clique-width, and as graph classes we consider classes defined by forbidding some specific graph $F$ as a minor, induced subgraph or subgraph, respectively. Our main focus is on treedepth for $F$-subgraph-free graphs of diameter at most~$d$ for some fixed integer $d$. We give classifications of boundedness of treedepth for $din {4,5,ldots}$ and partial classifications for $d=2$ and $d=3$.

Problem

• Determine if graph width becomes bounded under diameter constraints
• Study treedepth for F-subgraph-free graphs with fixed diameter
• Classify boundedness of treedepth for various diameter ranges

Innovation

• Analyze graph width changes with bounded diameter
• Focus on treedepth for F-subgraph-free graphs
• Classify boundedness for various diameter ranges