Barycentric Drawings of Periodic Graphs

  • Olaf Delgado-Friedrichs
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2912)


We study barycentric placement of vertices in periodic graphs of dimension 2 or higher. Barycentric placements exist for every connected periodic graph, are unique up to affine transformations, and provide a versatile tool not only in drawing, but also in computation. Example applications include symmetric convex drawing in dimension 2 as well as determining topological types of crystals and computing their ideal symmetry groups.


  1. [BK79]
    Bachem, A., Kannan, R.: Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM Journal Computing 8, 499–507 (1979)MATHCrossRefMathSciNetGoogle Scholar
  2. [CHK84]
    Chung, S.J., Hahn, T., Klee, W.E.: Nomenclature and generation of three-periodic nets: the vector method. Acta Cryst. A40, 42–50 (1984)MathSciNetGoogle Scholar
  3. [CM91]
    Cohen, E., Megiddo, N.: Recognizing properties of periodic graphs. In: Applied geometry and discrete mathematics. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, pp. 135–146. Amer. Math. Soc., Providence (1991)Google Scholar
  4. [DDH+99]
    Delgado Friedrichs, O., Dress, A.W.M., Huson, D.H., Klinowski, J., Mackay, A.L.: Systematic enumeration of crystalline networks. Nature 400, 644–647 (1999)CrossRefGoogle Scholar
  5. [Del01]
    Delgado-Friedrichs, O.: Equilibrium placement of periodic graphs and tilings (2001) (submitted)Google Scholar
  6. [Ead84]
    Eades, P.: A heuristic for graph drawing. Congressus Numerantium 42, 149–160 (1984)MathSciNetGoogle Scholar
  7. [FR91]
    Fruchterman, T., Reingold, E.: Graph drawing by force-directed placement. Software—Practice and Experience 21(11), 1129–1164 (1991)CrossRefGoogle Scholar
  8. [Hah83]
    Hahn, T. (ed.): International Tables for Crystallography. A. D. Reidel Publishing Company, Dordrecht (1983)Google Scholar
  9. [Kle87]
    Klee, W.E.: The topology of crystal structures. Z. Kristallogr. 179, 67–76 (1987)MATHCrossRefMathSciNetGoogle Scholar
  10. [OB92]
    O’Keeffe, M., Brese, N.E.: Uninodal 4-connected 3d nets. I. Nets without 3- or 4-rings. Acta Cryst. A48, 663–669 (1992)Google Scholar
  11. [OEL+00]
    O’Keeffe, M., Eddaoudi, M., Li, H., Reineke, T., Yaghi, O.M.: Frameworks for extended solids: Geometrical design principles. J. Solid State Chem. 152(1), 3–20 (2000)CrossRefGoogle Scholar
  12. [Orl84]
    Orlin, J.B.: Some problems on dynamic/periodic graphs. In: Progress in combinatorial optimization (Waterloo, Ont., 1982), pp. 273–293. Academic Press, Toronto (1984)Google Scholar
  13. [RG96]
    Richter-Gebert, J.: Realization Spaces of Polytopes. Springer, Berlin (1996)Google Scholar
  14. [Sch80]
    Schwarzenberger, R.L.E.: n-dimensional crystallography. Research Notes in Mathematics, vol. 41. Pitman (Advanced Publishing Program), Boston (1980)MATHGoogle Scholar
  15. [Tho80]
    Thomassen, C.: Planarity and duality of finite and infinite graphs. Journal of Combinatorial Theory, Series B 29, 244–271 (1980)MATHCrossRefMathSciNetGoogle Scholar
  16. [TRR+97]
    Treacy, M.M.J., Randall, K.H., Rao, S., Perry, J.A., Chadi, D.J.: Enumeration of periodic tetrahedral frameworks. Z. Krist. 212, 768–791 (1997)CrossRefGoogle Scholar
  17. [Tut60]
    Tutte, W.T.: Convex representations of graphs. Proc. London Math. Soc. 10(3), 304–320 (1960)MATHCrossRefMathSciNetGoogle Scholar
  18. [Tut63]
    Tutte, W.T.: How to draw a graph. Proc. London Math. Soc. 13, 743–767 (1963)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Olaf Delgado-Friedrichs
    • 1
  1. 1.WSI Computer ScienceUniversity of TübingenTübingenGermany

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