Optimal Listing of Cycles and st-Paths in Undirected Graphs

TL;DR

This paper introduces the first optimal algorithms for listing all cycles and s-t paths in undirected graphs, achieving minimal total computational cost proportional to input size and output size.

Contribution

It presents the first optimal algorithms for enumerating all cycles and s-t paths in undirected graphs, reducing the problem to path listing and achieving optimal efficiency.

Findings

Achieves total running time proportional to input plus output size.

Provides the first optimal algorithms for cycle listing in undirected graphs.

Reduces cycle listing to an optimal s-t path listing problem.

Abstract

We present the first optimal algorithm for the classical problem of listing all the cycles in an undirected graph. We exploit their properties so that the total cost is the time taken to read the input graph plus the time to list the output, namely, the edges in each of the cycles. The algorithm uses a reduction to the problem of listing all the paths from a vertex s to a vertex t which we also solve optimally.