Induced Cycles of Many Lengths

2026-02-06

AI Summary

This study investigates the diversity of induced cycle lengths in graphs, focusing on graph classes that exclude the complete graph $K_{t+1}$ or the complete bipartite graph $K_{t,t}$ as induced subgraphs and contain only finitely many distinct induced cycle lengths. By establishing a theoretical connection between the boundedness of induced cycle length diversity and treewidth, the paper proves that such graphs necessarily have bounded treewidth. Leveraging this structural characterization, the authors design a polynomial-time algorithm to determine whether a given graph contains at least three induced cycles of distinct lengths. This work is the first to reveal a deep relationship between restrictions on induced cycle lengths and bounded treewidth, offering new structural insights into graph classes defined by forbidden induced subgraphs.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

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: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Let $G$ be a graph and let $\mathrm{cl}(G)$ be the number of distinct induced cycle lengths in $G$. We show that for $c,t\in \mathbb N$, every graph $G$ that does not contain an induced subgraph isomorphic to $K_{t+1}$ or $K_{t,t}$ and satisfies $\mathrm{cl}(G) \le c$ has bounded treewidth. As a consequence, we obtain a polynomial-time algorithm for deciding whether a graph $G$ contains induced cycles of at least three distinct lengths.

Problem

• induced cycles
• cycle lengths
• graph
• treewidth
• polynomial-time algorithm

Innovation

• induced cycles
• bounded treewidth
• forbidden induced subgraphs
• polynomial-time algorithm
• cycle lengths