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Degree-constrained orientations of embedded graphs


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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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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Correspondence to Jannik Matuschke.

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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).

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