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

TL;DR

The paper introduces a deterministic linear-time algorithm to find an odd cycle through two specified vertices in an undirected graph, generalizing to group-labeled graphs with elements of order at most 2.

Contribution

It presents a novel linear-time algorithm for detecting odd cycles through two vertices, extending to group-labeled graphs with specific algebraic properties.

Findings

Algorithm runs in linear time for the problem.

Can determine existence of two cycles with distinct labels through specified vertices.

Finds such cycles efficiently if they exist.

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 $\Gamma$ be any group in which every element is of order at most $2$. For a given $\Gamma$-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.