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Degree-constrained orientations of embedded graphs

Author

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  • Yann Disser

    (TU Berlin)

  • Jannik Matuschke

    (TU Berlin)

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 $$2^{2g}$$ 2 2 g , where $$g$$ 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 $$N\!P$$ N P -complete even for a fixed genus if only upper and lower bounds on the in-degrees are specified instead of exact values.

Suggested Citation

  • Yann Disser & Jannik Matuschke, 2016. "Degree-constrained orientations of embedded graphs," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 758-773, February.
  • Handle: RePEc:spr:jcomop:v:31:y:2016:i:2:d:10.1007_s10878-014-9786-1
    DOI: 10.1007/s10878-014-9786-1
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    References listed on IDEAS

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    1. Steffen Rebennack & Ashwin Arulselvan & Lily Elefteriadou & Panos M. Pardalos, 2010. "Complexity analysis for maximum flow problems with arc reversals," Journal of Combinatorial Optimization, Springer, vol. 19(2), pages 200-216, February.
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    Cited by:

    1. Zhang, Weijuan & Qian, Jianguo, 2023. "Flips on homologous orientations of surface graphs with prescribed forbidden facial cycles," Applied Mathematics and Computation, Elsevier, vol. 444(C).

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