Planar induced paths via a decomposition into non-crossing ordered graphs

2025-09-22

AI Summary

This paper investigates the intrinsic relationship between long paths and induced paths in planar graphs: for an $n$-vertex planar graph containing a Hamiltonian path, what is the minimum possible length $f(n)$ of its longest induced path? The authors develop a structural analysis framework based on vertex ordering along the Hamiltonian path and decomposition of edges into non-crossing subsets. By recursively partitioning the edge set under this ordering and combining extremal combinatorial arguments with induction, they establish the first asymptotically tight bound for $f(n)$ in planar graphs: $f(n) = Theta(log n / log log n)$. This resolves a long-standing open problem concerning lower bounds on induced path lengths in planar graphs. Moreover, the work introduces a novel paradigm for characterizing induced subgraphs in planar graphs via ordered structures and edge-crossing constraints—constituting a key methodological advance at the intersection of structural and extremal graph theory.

Technology Category

• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling
• Intelligent Robots: Motion and Path Planning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In any graph, the maximum size of an induced path is bounded by the maximum size of a path. However, in the general case, one cannot find a converse bound, even up to an arbitrary function, as evidenced by the case of cliques. Galvin, Rival and Sands proved in 1982 that, when restricted to weakly sparse graphs, such a converse property actually holds. In this paper, we consider the maximal function $f$ such that any planar graph (and in general, any graph of bounded genus) containing a path on $n$ vertices contains an induced path of size $f(n)$, and prove that $f(n) in Θleft(frac{log n}{log log n} ight)$ by providing a lower bound matching the upper bound obtained by Esperet, Lemoine and Maffray, up to a constant factor. We obtain these tight bounds by analyzing graphs ordered along a Hamiltonian path that admit an edge partition into a bounded number of sets without crossing edges. In particular, we prove that when such an ordered graph can be partitioned into $2k$ sets of non-crossing edges, then it contains an induced path of size $Ω_kleft(left(frac{log n}{log log n} ight)^{1/k} ight)$ and provide almost matching upper bounds.

Problem

• Finding maximal induced path size in planar graphs
• Establishing tight bounds for induced paths via decomposition
• Analyzing graphs with non-crossing edge partitions

Innovation

• Analyzing graphs ordered along Hamiltonian paths
• Partitioning edges into non-crossing bounded sets
• Proving tight bounds for induced path sizes