Distance Between α-Orientations of Plane Graphs by Facial Cycle Reversals

  • Wei Juan Zhang
  • Jian Guo QianEmail author
  • Fu Ji Zhang


Cycle reversal had been shown as a powerful method to deal with the relation among orientations of a graph since it preserves the out-degree of each vertex and the connectivity of the orientations. A facial cycle reversal on an orientation of a plane graph is an operation that reverses all the directions of the edges of a directed facial cycle. An orientation of a graph is called an α- orientation if each vertex admits a prescribed out-degree. In this paper, we give an explicit formula for the minimum number of the facial cycle reversals needed to transform one α-orientation into another for plane graphs.


α-Orientation facial cycle reversal distance plane graph 

MR(2010) Subject Classification

05C10 05C76 05C12 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors would like to thank the anonymous reviewers for their constructive comments and suggestions.


  1. [1]
    De Fraysseix, H., De Mendez, P. O.: On topological aspects of orientations. Discrete Math., 229, 57–72 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Disser, Y., Matuschke, J.: Degree-constrained orientations of embedded graphs. J. Comb. Optim., 31, 758–773 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Felsner, S.: Lattice structures from planar graphs. Electron. J. Combin., 11, #R15 (2004)MathSciNetzbMATHGoogle Scholar
  4. [4]
    Felsner, S., Zickfeld, F.: On the number of planar orientations with prescribed degrees. Electron. J. Combin., 15, #R77 (2008)MathSciNetzbMATHGoogle Scholar
  5. [5]
    Frank, A., Gyárfás, A.: How to orient the edges of a graph. Colloq. Math. Soc. János Bolyai, 18, 353–364 (1976)MathSciNetGoogle Scholar
  6. [6]
    Hakimi, S. L.: On the degrees of the vertices of a directed graph. J. Franklin Inst., 279(4), 290–308 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Hassin, R.: Maximum flow in (s, t) planar networks. Inf. Processing Letters, 13, 107–107 (1981)MathSciNetCrossRefGoogle Scholar
  8. [8]
    Knauer, K. B.: Partial orders on orientations via cycle flips [Ph.D thesis], Technische Universität Berlin, Berlin, 2007Google Scholar
  9. [9]
    Lam, P. C. B., Zhang, H. P.: A distributive lattice on the set of perfect matchings of a plane bipartite graph. Order, 20, 13–29 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Nakamoto, A., Ota, K., Tanuma, T.: Three-cycle reversions in oriented planar triangulations. Yokohama Math. J., 44, 123–139 (1997)MathSciNetzbMATHGoogle Scholar
  11. [11]
    Zhang, F. J., Guo, X. F., Chen, R. S.: Z-transformation graphs of perfect matchings of hexagonal systems. Discrete Math., 72, 405–415 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Zhang, H. P.: Direct sum of distributive lattices on the perfect matchings of a plane bipartite graph. Order, 27(2), 101–113 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Zhang, H. P., Lam, Peter C. B. Lam, Shiu, W. C.: Cell rotation graphs of strongly connected orientations of plane graphs with an application. Discrete Appl. Math., 130(3), 469–485 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Zhang, H. P., Lam, Peter C. B. Lam, Shiu, W. C.: Resonance graphs and a binary coding for the 1-factors of benzenoid systems. SIAM J. Discrete Math., 22(3), 971–984 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Zhang, H. P., Zhang, F. J.: Plane elementary bipartite graphs. Discrete Appl. Math., 105, 291–311 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Zhang, W. J., Qian, J. G., Zhang, F. J.: Flip-distance between a-orientations of graphs embedded on plane and sphere. J. Xiamen Univ. (Natural Science), to appearGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science (CAS), Chinese Mathematical Society (CAS) and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Wei Juan Zhang
    • 1
    • 2
  • Jian Guo Qian
    • 1
    Email author
  • Fu Ji Zhang
    • 1
  1. 1.School of Mathematical SciencesXiamen UniversityXiamenP. R. China
  2. 2.School of Mathematical SciencesXinjiang Normal UniversityUrumqiP. R. China

Personalised recommendations