Abstract
The Feedback problem is one of the classical \(\mathcal{NP}\)-hard problems. Given a graph with n vertices and m arcs, it asks for a subset of arcs whose deletion makes a graph acyclic. An equivalent is the Linear Ordering, where the vertices are ordered from 1 to n, and a feedback arc is an arc that is directed contrarily. Both problems have been studied intensely.
Here, we add a new point of view. We first derive properties of linear orderings, that can be established efficiently. Our main result are upper bounds on the cardinality of a minimum feedback arc set for graphs with degree at most 3 and 4. We prove that the bounds are at most n/3 and m/3, respectively, and show that both are tight.
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Hanauer, K., Brandenburg, F.J., Auer, C. (2013). Tight Upper Bounds for Minimum Feedback Arc Sets of Regular Graphs. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds) Graph-Theoretic Concepts in Computer Science. WG 2013. Lecture Notes in Computer Science, vol 8165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45043-3_26
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DOI: https://doi.org/10.1007/978-3-642-45043-3_26
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