## Abstract

We investigate the problem of orienting the edges of an embedded graph in such a way that the resulting digraph fulfills given in-degree specifications both for the vertices and for the faces of the embedding. This *primal-dual orientation problem* was first proposed by Frank for the case of planar graphs, in conjunction with the question for a good characterization of the existence of such orientations. We answer this question by showing that a feasible orientation of a planar embedding, if it exists, can be constructed by combining certain parts of a primally feasible orientation and a dually feasible orientation. For the general case of arbitrary embeddings, we show that the number of feasible orientations is bounded by \(2^{2g}\), where \(g\) is the genus of the embedding. Our proof also yields a fixed-parameter algorithm for determining all feasible orientations parameterized by the genus. In contrast to these positive results, however, we also show that the problem becomes \(N\!P\)-complete even for a fixed genus if only upper and lower bounds on the in-degrees are specified instead of exact values.

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## Notes

A

*good characterization*good characterization of a decision problem in the sense of Edmonds (1965) is a description of polynomially verifiable certificates for both yes- and no-instances of the problem.Personal communication at the Seminar of the Egerváry research group on combinatorial optimization, February 2010.

A simple cut is a cut whose edge set is minimal w.r.t. inclusion. In a connected graph, a cut is simple if and only if it splits the graph into two connected components.

The term “rigid” for edges that are oriented in an identical way in all feasible orientations was introduced by Felsner (2004).

Personal communication during the Colloquium on Combinatorics 2012.

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## Acknowledgments

We thank Kristóf Bérczi and Júlia Pap for introducing us to this interesting topic and providing many helpful suggestions. This work was supported by the Alexander von Humboldt Foundation, by the Berlin Mathematical School and by Deutsche Forschungsgemeinschaft as part of the Priority Program “Algorithm Engineering” (1307).

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## Additional information

An extended abstract of this article has appeared in the proceedings of the 23rd International Symposium on Algorithms and Computation (ISAAC 2012), Taipei, 2012 (Disser and Matuschke 2012).

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Disser, Y., Matuschke, J. Degree-constrained orientations of embedded graphs.
*J Comb Optim* **31**, 758–773 (2016). https://doi.org/10.1007/s10878-014-9786-1

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DOI: https://doi.org/10.1007/s10878-014-9786-1

### Keywords

- Graph orientation
- Graph embeddings
- Planar graphs
- Fixed-parameter tractability
- Complexity theory