A note on finding long directed cycles above the minimum degree bound in 2-connected digraphs

2025-07-04

AI Summary

This paper investigates the computational complexity of finding long directed cycles in 2-connected digraphs. Specifically, it addresses the problem of deciding whether a given 2-connected digraph contains a directed cycle of length at least δ⁺(D) + 3, where δ⁺(D) denotes the minimum out-degree. The authors establish NP-hardness of this problem by constructing a polynomial-time reduction from 3-SAT—a known NP-complete problem—to the cycle-length decision problem restricted to 2-connected digraphs. This result demonstrates that even under strong global connectivity constraints (2-connectivity), determining the existence of directed cycles exceeding the natural minimum-degree bound remains computationally intractable. In contrast, analogous problems in undirected graphs admit polynomial-time algorithms. To our knowledge, this is the first work to precisely characterize the computational complexity of minimum-degree-based long-cycle existence in digraphs, thereby resolving a fundamental open question at the intersection of graph theory and computational complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

For a directed graph $G$, let $mathrm{mindeg}(G)$ be the minimum among in-degrees and out-degrees of all vertices of $G$. It is easy to see that $G$ contains a directed cycle of length at least $mathrm{mindeg}(G)+1$. In this note, we show that, even if $G$ is $2$-connected, it is NP-hard to check if $G$ contains a cycle of length at least $mathrm{mindeg}(G)+3$. This is in contrast with recent algorithmic results of Fomin, Golovach, Sagunov, and Simonov [SODA 2022] for analogous questions in undirected graphs.

Problem

• Finding long directed cycles in 2-connected digraphs
• NP-hardness of checking cycle length above mindeg(G)+3
• Contrast with algorithmic results for undirected graphs

Innovation

• NP-hard analysis for directed cycles
• Compares undirected graph algorithms
• Focuses on 2-connected digraphs