Degree-Constrained Orientations of Embedded Graphs

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 2(2g), where g is the genus of the embedding, and all solutions can be determined within time O(2(2g)vertical bar E vertical bar(2) + vertical bar E vertical bar(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.