Abstract
We investigate the problem of orienting the edges of an embedded graph in such a way that the in-degrees of both the nodes and faces meet given values. We show that the number of feasible solutions is bounded by 22g, where g is the genus of the embedding, and all solutions can be determined within time \(\mathcal{O}(2^{2g}|E|^2 + |E|^3)\). In particular, for planar graphs the solution is unique if it exists, and in general the problem of finding a feasible orientation is fixed-parameter tractable in g. In sharp contrast to these results, we show that the problem becomes NP-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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Disser, Y., Matuschke, J. (2012). Degree-Constrained Orientations of Embedded Graphs. In: Chao, KM., Hsu, Ts., Lee, DT. (eds) Algorithms and Computation. ISAAC 2012. Lecture Notes in Computer Science, vol 7676. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35261-4_53
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DOI: https://doi.org/10.1007/978-3-642-35261-4_53
Publisher Name: Springer, Berlin, Heidelberg
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