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

TL;DR

This paper investigates the computational complexity of finding long directed cycles in 2-connected digraphs, showing that determining cycles significantly longer than the minimum degree bound is NP-hard, contrasting with known results for undirected graphs.

Contribution

It establishes the NP-hardness of detecting long directed cycles above the minimum degree bound in 2-connected digraphs, highlighting a complexity difference from undirected cases.

Findings

Checking for cycles of length at least 'mindeg+3' is NP-hard.

Existence of long cycles is easy to verify at the 'mindeg+1' level.

Contrasts with recent polynomial algorithms for undirected graphs.

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.