A Linear-Time Algorithm for Finding an Odd Cycle Through Two Specified Vertices
This solves a fundamental graph theory problem efficiently, providing a linear-time solution for a problem previously lacking such an algorithm.
The paper presents a deterministic linear-time algorithm for finding an odd cycle through two specified vertices in an undirected graph, generalized to any group where every element has order at most 2.
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.