Conflict-Free Graph Orientations with Parity Constraints

  • Sarah Cannon
  • Mashhood Ishaque
  • Csaba D. Tóth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7288)


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall (February 1993)Google Scholar
  2. 2.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge Univ. Press (2009)Google Scholar
  3. 3.
    Cannon, S., Ishaque, M., Tóth, C.D.: Conflict-free graph orientations with parity and degree constraints, arXiv:1203.3256 (2012) (manuscript)Google Scholar
  4. 4.
    Darmann, A., Pferschy, U., Schauer, J., Woeginger, G.J.: Paths, trees, and matchings under disjunctive constraints. Discrete Appl. Math. 159(16), 1726–1735 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Felsner, S., Fusy, É., Noy, M.: Asymptotic enumeration of orientations. Discrete Math. Theor. Comp. Sci. 12(2), 249–262 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Felsner, S., Zickfeld, F.: On the number of planar orientations with prescribed degrees. Electron. J. Comb. 15(1), article R77 (2008)Google Scholar
  7. 7.
    Frank, A.: On the orientaiton of graphs. J. Combin. Theor. B 28, 251–261 (1980)zbMATHCrossRefGoogle Scholar
  8. 8.
    Frank, A., Gyárfás, A.: How to orient the edges of a graph. Coll. Math. Soc. J. Bolyai 18, 353–364 (1976)Google Scholar
  9. 9.
    Frank, A., Jordán, T., Szigeti, Z.: An orientation theorem with parity conditions. Discrete Appl. Math. 115, 37–47 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Frank, A., Király, Z.: Graph orientations with edge-connection and parity constraints. Combinatorica 22, 47–70 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Frank, A., Tardos, É., Sebő, A.: Covering directed and odd cuts. Math Prog. Stud. 22, 99–112 (1984)zbMATHCrossRefGoogle Scholar
  12. 12.
    Hakimi, S.L.: On the degrees of the vertices of a directed graph. J. Franklin Inst. 279, 280–308 (1965)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Khanna, S., Naor, J., Shepherd, F.B.: Directed network design with orientation constraints. SIAM J. Discre. Math. 19, 245–257 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Lovász, L., Plummer, M.D.: Matching Theory. AMS Chelsea (2009)Google Scholar
  15. 15.
    Szabó, T., Welzl, E.: Unique sink orientations of cubes. In: Proc. 42nd FOCS, pp. 547–555. IEEE (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sarah Cannon
    • 1
    • 2
  • Mashhood Ishaque
    • 2
  • Csaba D. Tóth
    • 2
    • 3
  1. 1.Department of MathematicsTufts UniversityMedfordUSA
  2. 2.Department of Computer ScienceTufts UniversityMedfordUSA
  3. 3.Department of Mathematics and StatisticsUniversity of CalgaryCanada

Personalised recommendations