# Triangulating planar graphs while minimizing the maximum degree

## Abstract

In this paper we study the problem of triangulating a planar graph *G* while minimizing the maximum degree Δ(*G*′) of the resulting triangulated planar graph *G′*. It is shown that this problem is NP-complete. Worst-case lower bounds for Δ(*G*′) with respect to Δ(*G*) are given. We describe a linear algorithm to triangulate planar graphs, for which the maximum degree of the triangulated graph is only a constant larger than the lower bounds. Finally we show that triangulating one face while minimizing the maximum degree can be achieved in polynomial time. We use this algorithm to obtain a polynomial exact algorithm to triangulate the interior faces of an outerplanar graph while minimizing the maximum degree.

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## References

- [1]Booth, K.S., and G.S. Lueker, Testing for the consecutive ones property, interval graphs and graph planarity testing using PQ-tree algorithms,
*J. of Computer and System Sciences*13 (1976), pp. 335–379.Google Scholar - [2]Chiba, N., T. Yamanouchi and Nishizeki, Linear algorithms for convex drawings of planar graphs, In: J.A. Bondy and U.S.R. Murty (Eds.),
*Progress in Graph Theory*, Academic Press, Toronto, 1984, pp. 153–173.Google Scholar - [3]Eswaran, K.P., and R.E. Tarjan, Augmentation problems,
*SIAM J. Comput.*5 (1976), pp. 653–665.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]Fraysseix, H. de, J. Pach and R. Pollack, How to draw a planar graph on a grid,
*Combinatorica*10 (1990), pp. 41–51.Google Scholar - [6]Fraysseix, H. de, and P. Rosenstiehl, A depth first characterization of planarity,
*Annals of Discrete Math.*13 (1982), pp. 75–80.Google Scholar - [7]Haandel, F. van,
*Straight Line Embeddings on the Grid*, Dept. of Comp. Science, Report no. INF/SCR-91-19, Utrecht University, 1991.Google Scholar - [8]Hopcroft, J., and R.E. Tarjan, Efficient planarity testing,
*J. ACM*21 (1974), pp. 549–568.Google Scholar - [9]Hsu, T., and V. Ramachandran, A linear time algorithm for triconnectivity augmentation, in:
*Proc. 32th Annnual IEEE Symp. on Found. on Comp. Science*, Porto Rico, 1991.Google Scholar - [10]Hsu, T., and V. Ramachandran,
*On Finding a Smallest Augmentation to Biconnect a Graph*, Computer Science Dept., University of Texas at Austin, Texas, Tech. Rep. TR-91-12, 1991.Google Scholar - [11]Kant, G.,
*Optimal Linear Planar Augmentation Algorithms for Outerplanar Graphs*, Techn. Rep. RUU-CS-91-47, Dept. of Computer Science, Utrecht University, 1991.Google Scholar - [12]Kant, G.,
*A Linear Implementation of De Fraysseix' Grid Drawing Algorithm*, Manuscript, Dept. of Comp. Science, Utrecht University, 1988.Google Scholar - [13]Kant, G., and H.L. Bodlaender, Planar graph augmentation problems, Extended Abstract in: F. Dehne, J.-R. Sack and N. Santoro (Eds.),
*Proc. 2nd Workshop on Data Structures and Algorithms*, Lecture Notes in Comp. Science 519, Springer-Verlag, Berlin/Heidelberg, 1991, pp. 286–298.Google Scholar - [14]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 - [15]Rosenthal, A., and A. Goldner, Smallest augmentations to biconnect a graph,
*SIAM J. Comput.*6 (1977), pp. 55–66.Google Scholar - [16]Schnyder, W., Embedding planar graphs on the grid, in:
*Proc. 1st Annual ACM-SIAM Symp. on Discr. Alg.*, San Francisco, 1990, pp. 138–147.Google Scholar - [17]Tutte, W.T., Convex representations of graphs,
*Proc. London Math. Soc.*, vol. 10 (1960), pp. 304–320.Google Scholar - [18]Woods, D.,
*Drawing Planar Graphs*, Ph.D. Dissertation, Computer Science Dept., Stanford University, CA, Tech. Rep. STAN-CS-82-943, 1982.Google Scholar