Finding All Bounded-Length Simple Cycles in a Directed Graph -- Revisited

TL;DR

This paper critically examines a 2021 algorithm for enumerating bounded-length cycles in directed graphs, identifies logical gaps, and proposes a corrected version that maintains computational efficiency.

Contribution

It provides concrete counterexamples to the original algorithm, analyzes the flaws, and introduces a corrected formulation that preserves its computational complexity.

Findings

Original algorithm fails to enumerate some valid cycles.

Logical gaps identified in the original proofs.

Corrected algorithm maintains $O((c + 1) imes k imes (n + e))$ complexity.

Abstract

In 2021, Gupta and Suzumura proposed a novel algorithm for enumerating all bounded-length simple cycles in directed graphs. In this work, we present concrete examples demonstrating that the proposed algorithm fails to enumerate certain valid cycles. Via these examples, we perform a detailed analysis pinpointing the specific points at which the proofs exhibit logical gaps. Furthermore, we propose a corrected formulation that resolves these issues while preserving the desirable property that the algorithm's computational complexity remains $O((c + 1) \cdot k \cdot (n + e))$ where $c$ is the number of simple cycles of a specified maximum length $k$, and $n$ and $e$ the number of graph nodes and edges respectively.