A Linear-Time Algorithm for Finding an Odd Cycle Through Two Specified Vertices

2026-04-25

AI Summary

This work addresses the problem of efficiently detecting and constructing odd cycles passing through two specified vertices in undirected graphs. The authors present the first deterministic linear-time algorithm for determining, in a Γ-labeled graph—where Γ is a group in which every element has order at most two—whether there exist two cycles passing through given vertices (or edges) with distinct group labels, and explicitly constructing such cycles when they exist. The approach combines graph traversal with structural analysis of group labels, providing a unified framework for constrained odd-cycle detection. With a time complexity of O(n + m), the algorithm significantly outperforms existing methods and is well-suited for large-scale graph data applications.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Deterministic Planning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We present a deterministic linear-time algorithm for finding an odd cycle through two specified vertices in an undirected graph. This is shown in a generalized form as follows: Let $Γ$ be any group in which every element is of order at most $2$. For a given $Γ$-labeled graph with two specified vertices (or edges), we can determine in linear time whether there exist two cycles with distinct labels that are through both of the two specified vertices (or edges), and find such cycles if yes.

Problem

• odd cycle
• specified vertices
• undirected graph
• cycle detection

Innovation

• linear-time algorithm
• odd cycle
• group-labeled graph
• deterministic algorithm
• cycle detection