Conflict-Free Graph Orientations with Parity Constraints
It is known that every multigraph with an even number of edges has an even orientation (i.e., all indegrees are even). We study parity constrained graph orientations under additional constraints. We consider two types of constraints for a multigraph G = (V,E): (1) an exact conflict constraint is an edge set C ⊆ E and a vertex v ∈ V such that C should not equal the set of incoming edges at v; (2) a subset conflict constraint is an edge set C ⊆ E and a vertex v ∈ V such that C should not be a subset of incoming edges at v. We show that it is NP-complete to decide whether G has an even orientation with exact or subset conflicts, for all conflict sets of size two or higher. We present efficient algorithms for computing parity constrained orientations with disjoint exact or subset conflict pairs.
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- 1.Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall (February 1993)Google Scholar
- 2.Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge Univ. Press (2009)Google Scholar
- 3.Cannon, S., Ishaque, M., Tóth, C.D.: Conflict-free graph orientations with parity and degree constraints, arXiv:1203.3256 (2012) (manuscript)Google Scholar
- 6.Felsner, S., Zickfeld, F.: On the number of planar orientations with prescribed degrees. Electron. J. Comb. 15(1), article R77 (2008)Google Scholar
- 8.Frank, A., Gyárfás, A.: How to orient the edges of a graph. Coll. Math. Soc. J. Bolyai 18, 353–364 (1976)Google Scholar
- 14.Lovász, L., Plummer, M.D.: Matching Theory. AMS Chelsea (2009)Google Scholar
- 15.Szabó, T., Welzl, E.: Unique sink orientations of cubes. In: Proc. 42nd FOCS, pp. 547–555. IEEE (2001)Google Scholar