GD 1994: Graph Drawing pp 119-130

# The polyhedral approach to the maximum planar subgraph problem: New chances for related problems

• Michael Jünger
• Petra Mutzel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 894)

## Abstract

In [JM94] we used a branch and cut algorithm in order to determine a maximum weight planar subgraph of a given graph. One of the motivations was to produce a nice drawing of a given graph by drawing the found maximum planar subgraph, and then augmenting this drawing by the removed edges. Our experiments indicate that drawing algorithms for planar graphs which require 2- or 3-connectivity, resp. degree-constraints, in addition to planarity often give “nicer” results. Thus we are led to the following problems:
1. (1)

Find a maximum planar subgraph with maximum degree d ∈ IN.

2. (2)

Augment a planar graph to a K-connected planar graph.

3. (3)

Find a maximum planar k-connected subgraph of a given k-connected graph.

4. (4)

Given a graph G, which is not necessarily planar and not necessarily k-connected, determine a new graph H by removing r edges and adding a edges such that the new graph H is planar, spanning, k-connected, each node v has degree at most D(v) and r+a is minimum.

Problems (1), (2) and (3) have been discussed in the literature, we argue that a solution to the newly defined problem (4) is most useful for our goal. For all four problems we give a polyhedral formulation by defining different linear objective functions over the same polytope which is the intersection of the planar subgraph polytope [JM93], the k-connected subgraph polytope [S92] and the degree-constrained subgraph polytope. We point out why we are confident that a branch and cut algorithm for the new problem will be an implementable and useful tool in automatic graph drawing.

### References

1. [BGHS92]
Beck, H.K.B., H.-P. Galil, R. Henkel, and E. Sedlmayr: Chemistry in circumstellar shells, I. Chromospheric radiation fields and dust formation in optically thin shells of M-giants. Astron. Astrophys. 265 (1992) 626–642Google Scholar
2. [C92]
Cimikowski, R.J.: An Empirical Analysis of Graph Planarization Heuristics. Computer Science Dept., Montana State Univ. (1992)Google Scholar
3. [E93]
4. [EM94]
5. [F92]
Frank, A.: Augmenting graphs to meet edge-connectivity requirements. SIAM J. Discr. Math. 5 (1992) 25–53Google Scholar
6. [H93]
Himsolt, M.: Konzeption und Implementierung von Grapheneditoren. Dissertation, Universität Passau (1993)Google Scholar
7. [HR91]
Hsu, T.-S. and V. Ramachandran: A linear time algorithm for triconnectivity augmentation. Proc. 32th Annual Symp. on Found. of Comp. Science, Puerto Rico (1991) 548–559Google Scholar
8. [HT74]
Hopcroft, J., and R.E. Tarjan: Efficient planarity testing. J. ACM 21 (1974) 549–568Google Scholar
9. [JM93]
Jünger, M. and P. Mutzel: Solving the Maximum Planar Subgraph Problem by Branch and Cut. Proceedings of the 3rd International Conference on Integer Programming and Combinatorial Optimization (IPCO 3), Erice (1993) 479–492Google Scholar
10. [JM94]
Jünger, M. and P. Mutzel: Maximum planar subgraphs and nice embeddings: Practical layout tools. to appear in Algorithmica, special issue on Graph Drawing, Edit. by G. Di Battista und R. Tamassia (1994)Google Scholar
11. [JTS89]
Jayakumar, R., K. Thulasiraman and M.N.S. Swamy: O(n 2) Algorithms for Graph Planarization. IEEE Trans. on Computer-aided Design 8 (1989) 257–267Google Scholar
12. [K92]
Kant, G.: An O(n 2) Maximal Planarization Algorithm based on PQ-trees. Technical Report, RUU-CS-92-03, Dept. of Computer Science, Utrecht University (1992)Google Scholar
13. [K93]
Kant, G.: Algorithms for Drawing Planar Graphs. Ph.D.-Thesis, Utrecht University (1993)Google Scholar
14. [M94a]
Mutzel, P.: The Maximum Planar Subgraph Problem. Dissertation, Universität Köln (1994)Google Scholar
15. [M94b]
Mutzel, P.: s-Chorded Cycle Graphs and their Relation to the Planar Subgraph Polytope. Technical Report No. 94-161, Angewandte Mathematik und Informatik, Universität zu Köln (1994)Google Scholar
16. [RG77]
Rosenthal, A. and A. Goldner: Smallest augmentation to biconnect a graph SIAM J. on Computing 6 (1977) 55–66Google Scholar
17. [S92]
Stoer, M.: Design of Survivable Networks. Lecture Notes in Mathematics, Springer-Verlag, Berlin (1992)Google Scholar
18. [STT81]
Sugiyama, K., S. Tagawa, and M. Toda: Methods for Visual Understanding of Hierarchical Systems. IEEE Trans. on Systems, Man and Cybernetics, SMC-11, 2 (1981) 109–125Google Scholar
19. [TBB88]
Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16 (1987) 421–444Google Scholar
20. [TBB88]
Tamassia, R., G. Di Battista, and C. Batini: Automatic graph drawing and readability of diagrams. IEEE Transactions on Systems, Man and Cybernetics 18 (1988) 61–79Google Scholar