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Free Edge Lengths in Plane Graphs

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Abstract

We study the impact of metric constraints on the realizability of planar graphs. Let G be a subgraph of a planar graph H (where H is the “host” of G). The graph G is free in H if for every choice of positive lengths for the edges of G, the host H has a planar straight-line embedding that realizes these lengths; and G is extrinsically free in H if all constraints on the edge lengths of G depend on G only, irrespective of additional edges of the host H. We characterize the planar graphs G that are free in every host H, \(G\subseteq H\), and all the planar graphs G that are extrinsically free in every host H, \(G\subseteq H\). The case of cycles \(G=C_k\) provides a new version of the celebrated carpenter’s rule problem. Even though cycles \(C_k\), \(k\ge 4\), are not extrinsically free in all triangulations, it turns out that “nondegenerate” edge lengths are always realizable, where the edge lengths are considered degenerate if the cycle can be flattened (into a line) in two different ways. Separating triangles, and separating cycles in general, play an important role in our arguments. We show that every star is free in a 4-connected triangulation (which has no separating triangle).

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References

  1. Angelini, P., Da Lozzo, G., Di Battista, G., Frati, F., Patrignani, M., Roselli, V.: Morphing planar graph drawings optimally. In: Proceedings of the 41st International Colloquium on Automata, Languages, and Programming (ICALP), Part I. Lecture Notes in Computer Science 8572, pp. 126–137. Springer, New York (2014)

  2. Angelini, P., Di Battista, G., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. ACM Transactions on Algorithms 11(4), Art No. 32 (2015)

  3. Cabello, S., Demaine, E.D., Rote, G.: Planar embeddings of graphs with specified edge lengths. J. Graph Algorithms Appl. 11(1), 259–276 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Connelly, R.: Generic global rigidity. Discrete Comput. Geom. 33(4), 549–563 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Connelly, R., Demaine, E.D., Rote, G.: Straightening polygonal arcs and convexifying polygonal cycles. Discrete Comput. Geom. 30(2), 205–239 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  7. Di Battista, G., Vismara, L.: Angles of planar triangular graphs. SIAM J. Discrete Math. 9(3), 349–359 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Eades, P., Wormald, N.C.: Fixed edge-length graph drawing is NP-hard. Discrete Appl. Math. 28, 111–134 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fáry, I.: On straight line representation of plane graphs. Acta. Sci. Math. Szeged 11, 229–233 (1948)

    MATH  Google Scholar 

  10. Frati, F., Patrignani, M.: A note on minimum-area straight-line drawings of planar graphs. In: 15th Graph Drawing. Lecture Notes in Computer Science 4875, pp. 339–344. Springer, New York (2008)

  11. Hong, S.-H., Nagamochi, H.: Convex drawings of graphs with non-convex boundary constraints. Discrete Appl. Math. 156, 2368–2380 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Jackson, B., Jordán, T.: Connected rigidity matroids and unique realizations of graphs. J. Comb. Theory Ser. B 94(1), 1–29 (2005)

    Article  MATH  Google Scholar 

  13. Jelínek, V., Kratochvíl, J., Rutter, I.: A Kuratowski-type theorem for planarity of partially embedded graphs. Comput. Geom. Theory Appl. 46(4), 466–492 (2013)

    Article  MATH  Google Scholar 

  14. Kurowski, M.: Planar straight-line drawing in an \(O(n)\times O(n)\) grid with angular resolution \(\Omega (1/n)\). In: Proceedings of the 31st Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM). Lecture Notes in Computer Science 3381, pp. 250–258. Springer, New York (2005)

  15. Lenhart, W.J., Whitesides, S.H.: Reconfiguring closed polygonal chains in euclidean \(d\)-space. Discrete Comput. Geom. 13, 123–140 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. Patrignani, M.: On extending a partial straight-line drawing. Found. Comput. Sci. 17(5), 1061–1069 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Sauer, N.W.: Distance sets of Urysohn metric spaces. Can. J. Math. 65, 222–240 (2013)

    Article  MATH  Google Scholar 

  18. Sauer, N.W.: Edge labelled graphs and metric spaces. In: Abstracts of the Erdős Centennial, p. 78. Budapest (2013)

  19. Schnyder, W.: Embedding planar graphs in the grid. In: Proceedings of the 1st ACM-SIAM Symposium on Discrete Algorithms, pp. 138–147. ACM, San Francisco (1990)

  20. Tutte, W.T.: How to draw a graph. Proc. Lond. Math. Soc. 3–13(1), 743–767 (1963)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The work on this problem started at the 10th Gremo Workshop on Open Problems (Bergün, GR, Switzerland) and continued at the MIT-Tufts Research Group on Computational Geometry. We thank all participants of these meetings for stimulating discussions.

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Authors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, MA, USA

    Zachary Abel & Sarah Eisenstat

  2. Cornell University, Ithaca, NY, USA

    Robert Connelly

  3. Columbia University, New York, NY, USA

    Radoslav Fulek

  4. École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Filip Morić

  5. University of Electro-Communications, Tokyo, Japan

    Yoshio Okamoto

  6. Freie Universität Berlin, Berlin, Germany

    Tibor Szabó

  7. California State University Northridge, Los Angeles, CA, USA

    Csaba D. Tóth

Authors
  1. Zachary Abel
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  2. Robert Connelly
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  3. Sarah Eisenstat
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  4. Radoslav Fulek
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  5. Filip Morić
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  6. Yoshio Okamoto
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  7. Tibor Szabó
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  8. Csaba D. Tóth
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Corresponding author

Correspondence to Csaba D. Tóth.

Additional information

Editor in Charge: János Pach

A preliminary version of this paper has appeared in the Proceedings of the 30th Annual Symposium on Computational Geometry (SoCG), 2014, ACM Press, pp. 426–435.

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Abel, Z., Connelly, R., Eisenstat, S. et al. Free Edge Lengths in Plane Graphs. Discrete Comput Geom 54, 259–289 (2015). https://doi.org/10.1007/s00454-015-9704-z

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  • DOI: https://doi.org/10.1007/s00454-015-9704-z

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