Transition Operations over Plane Trees

  • Torrie L. Nichols
  • Alexander Pilz
  • Csaba D. Tóth
  • Ahad N. Zehmakan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

The operation of transforming one spanning tree into another by replacing an edge has been considered widely, both for general and geometric graphs. For the latter, several variants have been studied (e.g., edge slides and edge rotations). In a transition graph on the set \(\mathcal {T}(S)\) of noncrossing straight-line spanning trees on a finite point set S in the plane, two spanning trees are connected by an edge if one can be transformed into the other by such an operation. We study bounds on the diameter of these graphs, and consider the various operations both on general point sets and sets in convex position. In addition, we address the problem variant where operations may be performed simultaneously. We prove new lower and upper bounds for the diameters of the corresponding transition graphs and pose open problems.

Notes

Acknowledgment

Key ideas for our results on simultaneous edge slides were discussed at the GWOP 2017 workshop in Pochtenalp, Switzerland. We thank all participants for the constructive atmosphere. Research by Nichols and Tóth was partially supported by the NSF awards CCF-1422311 and CCF-1423615. Pilz is supported by a Schrödinger fellowship of the Austrian Science Fund (FWF): J-3847-N35.

References

  1. 1.
    Aichholzer, O., Asinowski, A., Miltzow, T.: Disjoint compatibility graph of non-crossing matchings of points in convex position. Electron. J. Comb. 22, P1 (2015)MathSciNetMATHGoogle Scholar
  2. 2.
    Aichholzer, O., Aurenhammer, F., Huemer, C., Krasser, H.: Transforming spanning trees and pseudo-triangulations. Inf. Process. Lett. 97(1), 19–22 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Aichholzer, O., Aurenhammer, F., Hurtado, F.: Sequences of spanning trees and a fixed tree theorem. Comput. Geom. 21(1–2), 3–20 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Aichholzer, O., Reinhardt, K.: A quadratic distance bound on sliding between crossing-free spanning trees. Comput. Geom. 37(3), 155–161 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Akl, S.G., Islam, M.K., Meijer, H.: On planar path transformation. Inf. Process. Lett. 104(2), 59–64 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Aloupis, G., Barba, L., Langerman, S., Souvaine, D.L.: Bichromatic compatible matchings. Comput. Geom. 48(8), 622–633 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Avis, D., Fukuda, K.: Reverse search for enumeration. Discret. Appl. Math. 65(1–3), 21–46 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bose, P., Czyzowicz, J., Gao, Z., Morin, P., Wood, D.R.: Simultaneous diagonal flips in plane triangulations. J. Graph Theory 54(4), 307–330 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bose, P., Hurtado, F.: Flips in planar graphs. Comput. Geom. 42(1), 60–80 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bose, P., Lubiw, A., Pathak, V., Verdonschot, S.: Flipping edge-labelled triangulations. Comput. Geom. (2017, in Press)Google Scholar
  11. 11.
    Buchin, K., Razen, A., Uno, T., Wagner, U.: Transforming spanning trees: a lower bound. Comput. Geom. 42(8), 724–730 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cano, J., Díaz-Báñez, J.M., Huemer, C., Urrutia, J.: The edge rotation graph. Graphs Comb. 29(5), 1–13 (2013)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Cayley, A.: A theorem on trees. Q. J. Math. 23, 376–378 (1889)MATHGoogle Scholar
  14. 14.
    Chang, J.M., Wu, R.Y.: On the diameter of geometric path graphs of points in convex position. Inf. Process. Lett. 109(8), 409–413 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Faudree, R., Schelp, R., Lesniak, L., Gyárfás, A., Lehel, J.: On the rotation distance of graphs. Discret. Math. 126(1), 121–135 (1994)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Chartrand, G., Saba, F., Zou, H.B.: Edge rotations and distance between graphs. Cas. Pest. Math. 110, 87–91 (1985)MathSciNetMATHGoogle Scholar
  17. 17.
    Galtier, J., Hurtado, F., Noy, M., Pérennes, S., Urrutia, J.: Simultaneous edge flipping in triangulations. Int. J. Comput. Geom. Appl. 13(2), 113–134 (2003)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Goddard, W., Swart, H.C.: Distances between graphs under edge operations. Discret. Math. 161(1), 121–132 (1996)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hernando, M., Hurtado, F., Márquez, A., Mora, M., Noy, M.: Geometric tree graphs of points in convex position. Discret. Appl. Math. 93(1), 51–66 (1999)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Hoffmann, M., Sharir, M., Sheffer, A., Tóth, C.D., Welzl, E.: Counting plane graphs: flippability and its applications. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 524–535. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22300-6_44 CrossRefGoogle Scholar
  21. 21.
    Huemer, C., de Mier, A.: Lower bounds on the maximum number of non-crossing acyclic graphs. Europ. J. Comb. 48, 48–62 (2015)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Ishaque, M., Souvaine, D.L., Tóth, C.D.: Disjoint compatible geometric matchings. Discret. Comput. Geom. 49, 89–131 (2013)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kanj, I.A., Sedgwick, E., Xia, G.: Computing the flip distance between triangulations. Discret. Comput. Geom. 58(2), 313–344 (2017)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Keller, C., Perles, M.A.: Reconstruction of the geometric structure of a set of points in the plane from its geometric tree graph. Discret. Comput. Geom. 55(3), 610–637 (2016)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lubiw, A., Masárová, Z., Wagner, U.: A proof of the orbit conjecture for flipping edge-labelled triangulations. In: Proceedings of the 33rd Symposium on Computational Geometry (SoCG 2017). LIPIcs, vol. 77, pp. 49:1–49:15. Schloss Dagstuhl (2017)Google Scholar
  26. 26.
    Lubiw, A., Pathak, V.: Flip distance between two triangulations of a point set is NP-complete. Comput. Geom. 49, 17–23 (2015)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Oxley, J.G.: Matroid Theory. Oxford University Press, Oxford (1993)MATHGoogle Scholar
  28. 28.
    Pilz, A.: Flip distance between triangulations of a planar point set is APX-hard. Comput. Geom. 47(5), 589–604 (2014)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Pournin, L.: A combinatorial method to find sharp lower bounds on flip distances. In: Proceedings of the 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), pp. 1–12 (2013)Google Scholar
  30. 30.
    Sleator, D.D., Tarjan, R.E., Thurston, W.P.: Rotation distance, triangulations, and hyperbolic geometry. J. Am. Math. Soc. 1, 647–681 (1988)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Wu, R.Y., Chang, J.M., Pai, K.J., Wang, Y.L.: Amortized efficiency of generating planar paths in convex position. Theor. Comput. Sci. 412(35), 4504–4512 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Torrie L. Nichols
    • 1
  • Alexander Pilz
    • 2
  • Csaba D. Tóth
    • 1
  • Ahad N. Zehmakan
    • 2
  1. 1.California State University NorthridgeLos AngelesUSA
  2. 2.Department of Computer ScienceETH ZürichZürichSwitzerland

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