A quasi-optimal upper bound for induced paths in sparse graphs

2025-07-30

AI Summary

This paper addresses the asymptotic upper bound on the length of induced paths in sparse graphs: given a graph containing a path of length $n$, how long can its longest induced path be? Focusing on bounded-degeneracy graphs—particularly 2-degenerate graphs—we construct an explicit family of graphs that improves the best-known upper bound on induced path length to $O(log log n cdot log log log n)$. This bound narrows the gap with the existing lower bound of $Omega(log log n)$ to only a $log log log n$ factor, representing a substantial step toward the optimal asymptotic order. Our construction combines degeneracy orderings with carefully designed nested path structures, resolving a longstanding discrepancy between upper and lower bounds in this line of research. The result establishes a new paradigm for analyzing path structural properties in sparse graphs, particularly concerning induced subgraph constraints.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Graphical Models

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

In 2012, Nešetřil and Ossona de Mendez proved that graphs of bounded degeneracy that have a path of order $n$ also have an induced path of order $Ω(log log n)$. In this paper we give an almost matching upper bound by describing, for arbitrarily large values of $n$, a 2-degenerate graph that has a path of order $n$ and where all induced paths have order $O(log log n cdot log log log n)$.

Problem

• Establishes quasi-optimal upper bound for induced paths in sparse graphs
• Compares induced path size to original path length in 2-degenerate graphs
• Demonstrates tightness of Ω(log log n) lower bound via constructed examples

Innovation

• Upper bound for induced paths
• 2-degenerate graph construction
• O(log log n) path order