# Planar graph augmentation problems

Conference paper

First Online:

## Abstract

In this paper we investigate the problem of adding a minimum number of edges to a planar graph in such a way that the resulting graph is biconnected and still planar. It is shown that this problem is NP-complete. We present an approximation algorithm for this planar biconnectivity augmentation problem that has performance ratio 3/2 and uses *O*(*n*^{2} log *n*) time. An *O*(*n*^{3}) approximation algorithm with performance ratio 5/4 is presented to make a biconnected planar graph triconnected by adding edges without losing planarity.

## Keywords

Planar Graph Performance Ratio Outgoing Edge Outerplanar Graph Matching Edge
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.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Di Battista, G., and R. Tamassia, Incremental planarity testing,
*Proc. 30th Annual IEEE Symp. on Found. on Comp. Science*, North Carolina, 1989, pp. 436–441.Google Scholar - [2]Eswaran, K.P., and R.E. Tarjan, Augmentation problems,
*SIAM J. Comput.*5 (1976), pp. 653–665.Google Scholar - [3]Frederickson, G.N., and J. Ja'Ja, Approximation algorithms for several graph augmentation problems,
*SIAM J. Comput.*10 (1981), pp. 270–283.Google Scholar - [4]Frank, A., Augmenting graphs to meet edge-connectivity requirements,
*Proc. 31th Annual IEEE Symp. on Found. on Comp. Science*, St. Louis, 1990, pp. 708–718.Google Scholar - [5]Gabow, H.N., Data structures for weighted matching and nearest common ancestors with linking, in:
*Proc. 1st Annual ACM-SIAM Symp. on Discrete Algorithms*, San Fransisco (1990), pp. 434–443.Google Scholar - [6]Harary, F.,
*Graph Theory*, Addison-Wesley Publ. Comp., Reading, Mass., 1969.Google Scholar - [7]Kant, G.,
*Optimal Linear Planar Augmentation Algorithms for Outerplanar Graphs*, in preparation.Google Scholar - [8]Micali, S., and V.V. Vazirani, An
*O*(\(\sqrt V\)·.*E*) algorithm for finding maximum matching in general graphs, in:*Proc. 21st Annual IEEE Symp. Foundations of Computer Science*, Syracuse (1980), pp. 17–27.Google Scholar - [9]Naor, D., D. Gusfield and C. Martel, A fast algorithm for optimally increasing the edge-connectivity,
*Proc. 31st Annual IEEE Symp. on Found. of Comp. Science*, St. Louis, 1990, pp. 698–707.Google Scholar - [10]Read, R.C., A new method for drawing a graph given the cyclic order of the edges at each vertex,
*Congr. Numer. 56*(1987), pp. 31–44.Google Scholar - [11]Rosenthal, A., and A. Goldner, Smallest augmentations to biconnect a graph,
*SIAM J. Comput.*6 (1977), pp. 55–66.Google Scholar - [12]Shiloach, Y., Another look at the degree constrained subgraph problem,
*Inf. Proc. Lett.*12 (1981), pp. 89–92.Google Scholar - [13]Tutte, W.T., Convex representations of graphs,
*Proc. London Math. Soc.*, vol. 10 (1960), pp. 304–320.Google Scholar - [14]Woods, D.,
*Drawing Planar Graphs*, Ph.D. Dissertation, Computer Science Dept., Stanford University, CA, Tech. Rep. STAN-CS-82-943, 1982.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1991