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arXiv · 2606.29336

An FPT algorithm for cycle rank on semi-complete digraphs

Seokbeom Kim·O-joung Kwon·Myounghwan Lee
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Abstract

Cycle rank is a depth parameter for digraphs introduced by Eggan in 1963. Gruber (DMTCS 2012) and Giannopoulou, Hunter, and Thilikos (DAM 2012) asked whether the problem of determining if a given digraph has cycle rank at most $w$ is fixed-parameter tractable parameterized by $w$. We provide such algorithms for semi-complete digraphs, and for digraphs of bounded directed clique-width. Specifically, we show that given an $n$-vertex semi-complete digraph $G$ and an integer $w$, one can in time $\mathcal{O}(9^{(w+1)4^{w+2}} \cdot n^2)$ determine whether $G$ has cycle rank at most $w$. The proof is reduced to the case of bounded directed clique-width, and we then show that given an $n$-vertex digraph $G$ with a directed clique-width $k$-expression and an integer $w$, one can in time $\mathcal{O}(9^{(w+1) 4^k} \cdot n)$ determine whether $G$ has cycle rank at most $w$. Additionally, we consider the \textsc{Minimum Feedback Arc Set} problem on semi-complete digraphs, and show that it can be solved in time $n^{\mathcal{O}(w)}$, where $w$ is the cycle rank of the given semi-complete digraph.

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Seokbeom Kim, O-joung Kwon, Myounghwan Lee. 2026-06-28. An FPT algorithm for cycle rank on semi-complete digraphs. https://arxiv.org/abs/2606.29336

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