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Regular Augmentation of Planar Graphs


In this paper, we study the problem of augmenting a planar graph such that it becomes \(k\)-regular, \(c\)-connected and remains planar, either in the sense that the augmented graph is planar, or in the sense that the input graph has a fixed (topological) planar embedding that can be extended to a planar embedding of the augmented graph. We fully classify the complexity of this problem for all values of \(k\) and \(c\) in both, the variable embedding and the fixed embedding case. For \(k \le 2\) all problems are simple and for \(k \ge 4\) all problems are NP-complete. Our main results are efficient algorithms (with running time \(O(n^{1.5}))\) for deciding the existence of a \(c\)-connected, 3-regular augmentation of a graph with a fixed planar embedding for \(c=0,1,2\) and a corresponding hardness result for \(c=3\). The algorithms are such that for yes-instances a corresponding augmentation can be constructed in the same running time.

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  1. Abellanas, M., García, A., Hurtado, F., Tejel, J., Urrutia, J.: Augmenting the connectivity of geometric graphs. Comput. Geom. Theory Appl. 40(3), 220–230 (2008)

    Article  MATH  Google Scholar 

  2. Al-Jubeh, M., Ishaque, M., Rédei, K., Souvaine, D.L., Tóth, C.D.: Tri-edge-connectivity augmentation for planar straight line graphs. In: Dong, Y., Du, D.-Z., Ibarra, O.H. (eds.) Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC’09), Volumes 5878 of Lecture Notes in Computer Science, pp. 902–912. Springer, Berlin (2009)

    Google Scholar 

  3. Angelini, P., Di Battista, G., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I. Testing planarity of partially embedded graphs. In: Proceedings 21st ACM-SIAM Symposium on Discrete Algorithms (SODA’10), pp. 202–221. SIAM (2010)

  4. David, D.W.: On Steinitz’s theorem concerning convex 3-polytopes and on some properties of planar graphs. In: Chartrand, G., Kapoor, S.F. (eds.) The Many Facets of Graph Theory, volume 110 of Lecture Notes in Mathematics, pp. 27–40. Springer, Berlin (1969)

    Google Scholar 

  5. de Berg, M., Khosravi, A.: Optimal binary space partitions for segments in the plane. Int. J. Comput. Geom. Appl. 22(3), 187–206 (2012)

    Article  MATH  Google Scholar 

  6. Dirac, G.A.: Some theorems on abstract graphs. Proc. Lond. Math. Soc. s3—-2, 69–81 (1952)

    Article  MathSciNet  Google Scholar 

  7. Eswaran, K.P., Tarjan, R.E.: Augmentation problems. SIAM J. Comput. 5(4), 653–665 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  8. Frederickson, G.N., JaJa, J.: Approximation algorithms for several graph augmentation problems. SIAM J. Comput. 10(2), 270–283 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gabow, H.N.: An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems. In: Proceedings of the 15th Annual ACM Symposium on Theory of Computing (STOC’83), pp. 448–456. ACM (1983)

  10. Hurtado, F., Tóth, C.D.: Plane geometric graph augmentation: a generic perspective. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, Algorithms and Combinatorics, vol. 29 (2012)

  11. Kant, G., Bodlaender, H.L.: Planar graph augmentation problems (extended abstract). In: Dehne, F.K.H.A., Sack, J.-R., Santoro, N. (eds.) Proceedings of the 2nd Workshop Algorithms and Data Structures (WADS’91), volume 519 of Lecture Notes in Computer Science, pp. 286–298. Springer, Berlin (1991)

  12. Knuth, D.E., Raghunathan, A.: The problem of compatible representatives. SIAM J. Discret. Math. 5(3), 422–427 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nagamochi, H., Ibaraki, T.: Graph connectivity and its augmentation: applications of MA orderings. Discret. Appl. Math. 1–3, 447–472 (2002)

    Article  MathSciNet  Google Scholar 

  15. Pilz, A.: Augmentability to cubic graphs. In: Proceedings of the 28th European Workshop on Computational Geometry (EuroCG’12), pp. 29–32 (2012)

  16. Rutter, I., Wolff, A.: Augmenting the connectivity of planar and geometric graphs. J. Graph Algorithms Appl. 16(2), 599–628 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tóth, C.D.: Connectivity augmentation in plane straight line graphs. Electron. Notes Discret. Math. 31, 49–52 (2008)

    Article  Google Scholar 

  18. Watanabe, T., Nakamura, A.: Edge-connectivity augmentation problems. J. Comput. Syst. Sci. 35(1), 96–144 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  19. Whitney, H.: Congruent graphs and the connectivity of graphs. Am. J. Math. 43, 150–168 (1932)

    Article  MathSciNet  Google Scholar 

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We thank an anonymous referee for several suggestions that helped us to simplify and shorten the algorithm in Sect. 2 and the connectivity proofs in Sect. 7.

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Correspondence to Ignaz Rutter.

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Partially supported by the DFG under grant WA 654/15 within the Priority Programme “Algorithm Engineering”.

A preliminary version of this paper has appeared as T. Hartmann, J. Rollin, I. Rutter, Cubic Augmentation of Planar Graphs, In Proceedings of the 23rd International Symposium on Algorithms and Computation (ISAAC’12), pages 402–412, volume 7676 of LNCS, 2013.

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Hartmann, T., Rollin, J. & Rutter, I. Regular Augmentation of Planar Graphs. Algorithmica 73, 306–370 (2015).

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  • Regular planar graphs
  • Graph augmentation
  • Complexity
  • Efficient algorithms
  • Connectivity