Abstract
For a multigraph G = (V, E), let s ∈ V be a designated vertex which has an even degree, and let λ G (V − s) denote min{c G(X) | Ø ≠ X ⊂ V − s}, where c G(X) denotes the size of cut X. Splitting two adjacent edges (s, u) and (s, v) means deleting these edges and adding a new edge (u, v). For an integer k, splitting two edges e 1 and e 2 incident to s is called (k, s)-feasible if λG′(V − s) ≥ k holds in the resulting graph G′. In this paper, we prove that, for a planar graph G and an even k or k = 3 with k ≤ λ G (V − s), there exists a complete (k, s)-feasible splitting at s such that the resulting graph G′ is still planar, and present an O(n 3 log n) time algorithm for finding such a splitting, where n = |V|. However, for every odd k ≥ 5, there is a planar graph G with a vertex s which has no complete (k, s)-feasible and planarity-preserving splitting. As an application of this result, we show that for an outerplanar graph G and an even integer k the problem of optimally augmenting G to a k-edge-connected planar graph can be solved in O(n 3 log n) time.
Similar content being viewed by others
References
J. Bang-Jensen, H.N. Gabow, T. Jordán, and Z. Szigeti, “Edge-connectivity augmentation with partition constraints,” SIAM J. Disc. Math., vol. 12, pp. 160–207, 1999.
J. Bang-Jensen and T. Jordán, “Edge-connectivity augmentation preserving simplicity,” SIAM J. Disc. Math., vol. 11, pp. 603–623, 1998.
G.-R. Cai and Y.-G. Sun, “The minimum augmentation of any graph to k-edge-connected graph,” Networks, vol. 19, pp. 151–172, 1989.
K.P. Eswaran and R.E. Tarjan, “Augmentation problems,” SIAM J. Computing, vol. 5, pp. 653–665, 1976.
S. Fialko and P. Mutzel, “A new approximation algorithm for the planar augmentation problem,” in Proc. of 9th Annual ACM-SIAM Symposium on Discrete Algorithms, 1998, pp. 260–269.
A. Frank, “Augmenting graphs to meet edge-connectivity requirements,” SIAM J. Discrete Mathematics, vol. 5, pp. 25–53, 1992.
A. Frank, T. Ibaraki, and H. Nagamochi, “On sparse subgraphs preserving connectivity properties,” J. Graph Theory, vol. 17, pp. 275–281, 1993.
H.N. Gabow, “Efficient splitting off algorithms for graphs,” in Proc. 26thACMSymposium on Theory of Computing, 1994, pp. 696–705.
Z. Galil and G.F. Italiano, “Reducing edge connectivity to vertex connectivity,” SIGACT News, vol. 22, pp. 57–61, 1991.
T. Jordán, “Two NP-complete augmentation problems,” Odense University Preprints no. 8, 1997.
T. Jordán, “Edge-splitting problems with demands,” Lecture Notes in Computer Science, 1610, Springer-Verlag, 7th Conference on Integer Programming and Combinatorial Optimization, 1999, pp. 273–288.
G. Kant, “Algorithms for drawing planar graphs,” Ph.D. Thesis, Dept. of Computer Science, Utrecht University, 1993.
G. Kant, “Augmenting outerplanar graphs,” J. Algorithms, vol. 21, pp. 1–25, 1996.
G. Kant and H.L. Bodlaender, “Planar graph augmentation problems,” Lecture Notes in Compter Science, 621, Springer-Verlag, 3rd Scand. Workshop on Algorithm Theory, pp. 258–271, 1992.
L. Lovász, Combinatorial Problems and Exercises, North-Holland, 1979.
H. Nagamochi and T. Ibaraki, “A linear time algorithm for computing 3-edge-connected components in multigraphs,” J. of Japan Society for Industrial and Applied Mathematics, vol. 9, pp. 163–180, 1992.
H. Nagamochi and T. Ibaraki, “Deterministic Õ (nm) time edge-splitting in undirected graphs,” J. Combinatorial Optimization, vol. 1, pp. 5–46, 1997.
H. Nagamochi and T. Ibaraki, “A note on minimizing submodular functions,” Information Processing Letters, vol. 67, pp. 239–244, 1998.
H. Nagamochi, S. Nakamura, and T. Ibaraki, “A simplified Õ (nm) time edge-splitting algorithm in undirected graphs,” Algorithmica, vol. 26, pp. 56–67, 2000.
H. Nagamochi, K. Nishimura, and T. Ibaraki, “Computing all small cuts in undirected networks,” SIAM J. Discrete Mathematics, vol. 10, pp. 469–481, 1997.
R.E. Tarjan, “Depth-first search and linear graph algorithms,” SIAM J. Computing, vol. 1, pp. 146–160, 1972.
T. Watanabe and A. Nakamura, “Edge-connectivity augmentation problems,” J. Comp. System Sci., vol. 35, pp. 96–144, 1987.
T. Watanabe, S. Taoka, and K. Onaga, “A linear-time algorithm for computing all 3-edge-components of a multigraph,” Trans. Inst. Electron. Inform. Comm. Eng. Jap., vol. E75-A, pp. 410–424, 1992.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nagamochi, H., Eades, P. An Edge-Splitting Algorithm in Planar Graphs. Journal of Combinatorial Optimization 7, 137–159 (2003). https://doi.org/10.1023/A:1024470929537
Issue Date:
DOI: https://doi.org/10.1023/A:1024470929537