Realization of Simply Connected Polygonal Linkages and Recognition of Unit Disk Contact Trees

  • Clinton Bowen
  • Stephane Durocher
  • Maarten Löffler
  • Anika Rounds
  • André Schulz
  • Csaba D. TóthEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)


We wish to decide whether a simply connected flexible polygonal structure can be realized in Euclidean space. Two models are considered: polygonal linkages (body-and-joint framework) and contact graphs of unit disks in the plane. (1) We show that it is strongly NP-hard to decide whether a given polygonal linkage is realizable in the plane when the bodies are convex polygons and their contact graph is a tree; the problem is weakly NP-hard already for a chain of rectangles, but efficiently decidable for a chain of triangles hinged at distinct vertices. (2) We also show that it is strongly NP-hard to decide whether a given tree is the contact graph of interior-disjoint unit disks in the plane.


Unit Disk Boolean Formula Regular Hexagon Fixed Orientation Contact Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Our results in Sect. 4 were developed at the First International Workshop on Drawing Algorithms for Networks of Changing Entities (DANCE 2014), held in Langbroek, the Netherlands, and supported by the NWO project 639.023.208. Research by Rounds and Tóth was supported in part by the NSF awards CCF-1422311 and CCF-1423615. Research by Durocher was supported in part by NSERC.


  1. 1.
    Alt, H., Knauer, C., Rote, G., Whitesides, S.: On the complexity of the linkage reconfiguration problem. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs, vol. 342, Contemporary Mathematics, pp. 1–14. AMS, Providence (2004)Google Scholar
  2. 2.
    Ballinger, B., Charlton, D., Demaine, E.D., Demaine, M.L., Iacono, J., Liu, C.-H., Poon, S.-H.: Minimal locked trees. In: Dehne, F., Gavrilova, M., Sack, J.-R., Tóth, C.D. (eds.) WADS 2009. LNCS, vol. 5664, pp. 61–73. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  3. 3.
    Biedl, T., Kant, G.: A better heuristic for orthogonal graph drawings. Comput. Geom. 9(3), 159–180 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bhatt, S.N., Cosmadakis, S.S.: The complexity of minimizing wire lengths in VLSI layouts. Inform. Process. Lett. 25(4), 263–267 (1987)zbMATHCrossRefGoogle Scholar
  5. 5.
    Breu, H., Kirkpatrick, D.G.: On the complexity of recognizing intersection and touching graphs of discs. In: Brandenburd, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 88–98. Spinger, Heidelberg (1996)Google Scholar
  6. 6.
    Breu, H., Kirkpatrick, D.G.: Unit disk graph recognition is NP-hard. Comput. Geom. 9, 3–24 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Cabello, S., Demaine, E.D., Rote, G.: Planar embeddings of graphs with specified edge lengths. J. Graph Alg. Appl. 11(1), 259–276 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Cheong, J.-S., van der Stappen, A.F., Goldberg, K., Overmars, M.H., Rimon, E.: Immobilizing hinged polygons. Int. J. Comput. Geom. Appl. 17(1), 45–70 (2007)zbMATHCrossRefGoogle Scholar
  9. 9.
    Connelly, R., Demaine, E.D.: Geometry and topology of polygonal linkages. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, ch. 9, pp. 197–218. CRC, Boca Raton (2004)Google Scholar
  10. 10.
    Connelly, R., Demaine, E.D., Rote, G.: Straightening polygonal arcs and convexifying polygonal cycles. Discrete Comput. Geom. 30(2), 205–239 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Connelly, R., Demaine, E.D., Demaine, M.L., Fekete, S.P., Langerman, S., Mitchell, J.S.B., Ribó, A., Rote, G.: Locked and unlocked chains of planar shapes. Discrete Comput. Geom. 44(2), 439–462 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Demaine, E.D., Eppstein, D., Erickson, J., Hart, G.W., O’Rourke, J.: Vertex-unfoldings of simplicial manifolds. In: 18th Sympos. on Comput. Geom., pp. 237–243. ACM Press, New York (2002)Google Scholar
  13. 13.
    Di Battista, G., Vismara, L.: Angles of planar triangular graphs. SIAM J. Discrete Math. 9(3), 349–359 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, Upper Saddle River (1999)zbMATHGoogle Scholar
  15. 15.
    Eades, P., Whitesides, S.: The realization problem for Euclidean minimum spanning trees is NP-hard. Algorithmica 16(1), 60–82 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Eades, P., Wormald, N.C.: Fixed edge-length graph drawing is NP-hard. Discrete Appl. Math. 28, 111–134 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Fekete, S.P., Houle, M.E., Whitesides, S.: The wobbly logic engine: Proving hardness of non-rigid geometric graph representation problems. In: Di Battista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 272–283. Springer, Heidelberg (1997)Google Scholar
  18. 18.
    Gregori, A.: Unit-length embedding of binary trees on a square grid. Inform. Process. Lett. 31, 167–173 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Hliněný, P.: Touching graphs of unit balls. In: Di Battista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 350–358. Springer, Heidelberg (1997)Google Scholar
  20. 20.
    Hliněný, P., Kratochvíl, J.: Representing graphs by disks and balls (a survey of recognition-complexity results). Discrete Math. 229(1–3), 101–124 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Klemz, B., Nöllenburg, M., Prutkin, R.: Recognizing weighted disk contact graphs. In: Di Giacomo, E., Lubiw, A. (eds.) GD 2015. LNCS, vol. 9411, pp. 433–446. LNCS, Spinger, Heidelberg (2015)Google Scholar
  22. 22.
    Reif, J.H.: Complexity of the mover’s problem and generalizations. In: 20th FoCS, pp. 421–427. IEEE, New York (1979)Google Scholar
  23. 23.
    Schaefer, M.: Realizability of graphs and linkages. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 461–482. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  24. 24.
    Streinu, I.: Pseudo-triangulations, rigidity and motion planning. Discrete Comput. Geom. 34(4), 587–635 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Whitney, H.: Congruent graphs and the connectivity of graphs. Amer. J. Math. 54, 150–168 (1932)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Clinton Bowen
    • 1
  • Stephane Durocher
    • 2
  • Maarten Löffler
    • 3
  • Anika Rounds
    • 4
  • André Schulz
    • 5
  • Csaba D. Tóth
    • 1
    • 4
    Email author
  1. 1.Department of MathematicsCalifornia State University NorthridgeLos AngelesUSA
  2. 2.Department of Computer ScienceUniversity of ManitobaWinnipegCanada
  3. 3.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands
  4. 4.Department of Computer ScienceTufts UniversityMedfordUSA
  5. 5.Theoretical Computer ScienceUniversity of HagenHagenGermany

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