Degree-constrained orientations of embedded graphs

Yann Disser () and Jannik Matuschke ()
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Yann Disser: TU Berlin
Jannik Matuschke: TU Berlin

Journal of Combinatorial Optimization, 2016, vol. 31, issue 2, No 18, 758-773

Abstract: 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.

Keywords: Graph orientation; Graph embeddings; Planar graphs; Fixed-parameter tractability; Complexity theory (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10878-014-9786-1

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