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arXiv · 2409.07680

Upper bounds on minimum size of feedback arc set of directed multigraphs with bounded degree

Gregory Gutin·Hui Lei·Anders Yeo·Yacong Zhou
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Abstract

An oriented multigraph is a directed multigraph without directed 2-cycles. Let ${\rm fas}(D)$ denote the minimum size of a feedback arc set in an oriented multigraph $D$. The degree of a vertex is the sum of its out- and in-degrees. In several papers, upper bounds for ${\rm fas}(D)$ were obtained for oriented multigraphs $D$ with maximum degree upper-bounded by a constant. Hanauer (2017) conjectured that ${\rm fas}(D)\le 2.5n/3$ for every oriented multigraph $D$ with $n$ vertices and maximum degree at most 5. We prove a strengthening of the conjecture: ${\rm fas}(D)\le m/3$ holds for every oriented multigraph $D$ with $m$ arcs and maximum degree at most 5. This bound is tight and improves a bound of Berger and Shor (1990,1997). It would be interesting to determine $c$ such that ${\rm fas}(D)\le cn$ for every oriented multigraph $D$ with $n$ vertices and maximum degree at most 5 such that the bound is tight. We show that $\frac{5}{7}\le c \le \frac{24}{29} < \frac{2.5}{3}$.

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Gregory Gutin, Hui Lei, Anders Yeo, Yacong Zhou. 2024-09-12. Upper bounds on minimum size of feedback arc set of directed multigraphs with bounded degree. https://arxiv.org/abs/2409.07680

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