Minimum Weight Connectivity Augmentation for Planar Straight-Line Graphs

  • Hugo A. Akitaya
  • Rajasekhar Inkulu
  • Torrie L. Nichols
  • Diane L. Souvaine
  • Csaba D. Tóth
  • Charles R. Winston
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10167)


We consider edge insertion and deletion operations that increase the connectivity of a given planar straight-line graph (PSLG), while minimizing the total edge length of the output. We show that every connected PSLG \(G=(V,E)\) in general position can be augmented to a 2-connected PSLG \((V,E\cup E^+)\) by adding new edges of total Euclidean length \(\Vert E^+\Vert \le 2\Vert E\Vert \), and this bound is the best possible. An optimal edge set \(E^+\) can be computed in \(O(|V|^4)\) time; however the problem becomes NP-hard when G is disconnected. Further, there is a sequence of edge insertions and deletions that transforms a connected PSLG \(G=(V,E)\) into a plane cycle \(G'=(V,E')\) such that \(\Vert E'\Vert \le 2\Vert \mathrm{MST}(V)\Vert \), and the graph remains connected with edge length below \(\Vert E\Vert +\Vert \mathrm{MST}(V)\Vert \) at all stages. These bounds are the best possible.


Span Tree Edge Incident Delaunay Triangulation Simple Cycle Unit Disk Graph 
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  1. 1.
    Abellanas, M., García, A., Hurtado, F., Tejel, J., Urrutia, J.: Augmenting the connectivity of geometric graphs. Comput. Geom. 40, 220–230 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Akitaya, H.A., Castello, J., Lahoda, Y., Rounds, A., Tóth, C.D.: Augmenting planar straight line graphs to 2-edge-connectivity. In: Di Giacomo, E., Lubiw, A. (eds.) GD 2015. LNCS, vol. 9411, pp. 563–564. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-27261-0_52 CrossRefGoogle Scholar
  3. 3.
    Akitaya, H.A., Inkulu, T., Nichols, T.L., Souvaine, D.L., Tóth, C.D., Winston, C.R.: Minimum weight connectivity augmentation for planar straight-line graphs. Preprint, arXiv:1612.04780 (2016)
  4. 4.
    Bespamyatnikh, B.: Computing homotopic shortest paths in the plane. J. Algorithms 49, 284–303 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    de Berg, M., Khosravi, A.: Optimal binary space partitions for segments in the plane. Int. J. Comput. Geom. Appl. 22, 187–205 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, D.Z., Wang, H.: A new algorithm for computing visibility graphs of polygonal obstacles in the plane. J. Comput. Geom. 6, 316–345 (2015)MathSciNetGoogle Scholar
  7. 7.
    Dobrev, S., Kranakis, E., Krizanc, D., Morales-Ponce, O., Stacho, L.: Approximating the edge length of 2-edge connected planar geometric graphs on a set of points. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 255–266. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-29344-3_22 CrossRefGoogle Scholar
  8. 8.
    Efrat, A., Kobourov, S.G., Lubiw, A.: Computing homotopic shortest paths efficiently. Comput. Geom. 35, 162–172 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Frank, A.: Connections in Combinatorial Optimization. Oxford Lecture Series in Mathematics and Its Applications, vol. 12. Oxford University Press, Oxford (2011)zbMATHGoogle Scholar
  10. 10.
    Frederickson, G.N., Ja’Ja’, J.: Approximation algorithms for several graph augmentation problems. SIAM J. Comput. 10, 270–283 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gutwenger, C., Mutzel, P., Zey, B.: Planar biconnectivity augmentation with fixed embedding. In: Fiala, J., Kratochvíl, J., Miller, M. (eds.) IWOCA 2009. LNCS, vol. 5874, pp. 289–300. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-10217-2_29 CrossRefGoogle Scholar
  12. 12.
    Hershberger, J., Snoeyink, J.: Computing minimum length paths of a given homotopy class. Comput. Geom. 4, 63–98 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hurtado, F., Tóth, C.D.: Plane geometric graph augmentation: a generic perspective. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 327–354. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  14. 14.
    Kant, G., Bodlaender, H.L.: Planar graph augmentation problems. In: Dehne, F., Sack, J.-R., Santoro, N. (eds.) WADS 1991. LNCS, vol. 519, pp. 286–298. Springer, Heidelberg (1991). doi: 10.1007/BFb0028270 CrossRefGoogle Scholar
  15. 15.
    Kortsarz, G., Nutov, Z.: A simplified 1.5-approximation algorithm for augmenting edge-connectivity of a graph from 1 to 2. ACM Trans. Algorithms 12, Article no. 23 (2016)Google Scholar
  16. 16.
    Kranakis, E., Krizanc, D., Ponce, O.M., Stacho, L.: Bounded length 2-edge augmentation of geometric planar graphs. Discret. Math. Algorithms Appl. 4(3) (2012). doi: 10.1142/S179383091250036X
  17. 17.
    Overmars, M.H., Welzl, E.: New methods for computing visibility graphs. In: Proceedings of 14th Symposium on Computational Geometry, pp. 164–171. ACM Press, New York (1988)Google Scholar
  18. 18.
    Rutter, I., Wolff, A.: Augmenting the connectivity of planar and geometric graphs. J. Graph Algorithms Appl. 16, 599–628 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tóth, D.C.: Connectivity augmentation in planar straight line graphs. Eur. J. Combin. 33, 408–425 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Végh, L.A.: Augmenting undirected node-connectivity by one. SIAM J. Discret. Math. 25, 695–718 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Hugo A. Akitaya
    • 1
  • Rajasekhar Inkulu
    • 2
  • Torrie L. Nichols
    • 3
  • Diane L. Souvaine
    • 1
  • Csaba D. Tóth
    • 1
    • 3
  • Charles R. Winston
    • 1
  1. 1.Tufts UniversityMedfordUSA
  2. 2.Indian Institute of Technology GuwahatiGuwahatiIndia
  3. 3.California State University NorthridgeLos AngelesUSA

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