Algorithmica

, Volume 16, Issue 1, pp 33–59 | Cite as

Maximum planar subgraphs and nice embeddings: Practical layout tools

  • M. Jünger
  • P. Mutzel
Article

Abstract

In automatic graph drawing a given graph has to be laid out in the plane, usually according to a number of topological and aesthetic constraints. Nice drawings for sparse nonplanar graphs can be achieved by determining a maximum planar subgraph and augmenting an embedding of this graph. This approach appears to be of limited value in practice, because the maximum planar subgraph problem is NP-hard.

We attack the maximum planar subgraph problem with a branch-and-cut technique which gives us quite good, and in many cases provably optimum, solutions for sparse graphs and very dense graphs. In the theoretical part of the paper, the polytope of all planar subgraphs of a graphG is defined and studied. All subgraphs of a graphG, which are subdivisions ofK5 orK3,3, turn out to define facets of this polytope. For cliques contained inG, the Euler inequalities turn out to be facet-defining for the planar subgraph polytope. Moreover, we introduce the subdivision inequalities,V2k inequalities, and the flower inequalities, all of which are facet-defining for the polytope. Furthermore, the composition of inequalities by 2-sums is investigated.

We also present computational experience with a branch-and-cut algorithm for the above problem. Our approach is based on an algorithm which searches for forbidden substructures in a graph that contains a subdivision ofK5 orK3,3. These structures give us inequalities which are used as cutting planes.

Finally, we try to convince the reader that the computation of maximum planar subgraphs is indeed a practical tool for finding nice embeddings by applying this method to graphs taken from the literature.

Key words

Maximum planar subgraph Planar subgraph polytope Facets Branch and cut 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BL]
    Booth, K. S., and G. S. Lueker, Testing for the consecutive ones property, interval graphs and graph planarity testing using PQ-tree algorithms,Journal of Computer and System Sciences 13 (1976), 335–379.Google Scholar
  2. [CHT]
    Cai, J., X. Han, and R. E. Tarjan, An O(m logn)-time algorithm for the maximal planar subgraph problem,SIAM Journal on Computing 22 (1993), 1142–1162.Google Scholar
  3. [C]
    Cimikowski, R. J., An empirical analysis of graph planarization heuristics, unpublished manuscript, Computer Science Dept, Montana State University (1992).Google Scholar
  4. [EFG]
    Eades P., L. R. Foulds, and J. W. Giffin. An efficient heuristic for identifying a maximum weight planar subgraph,Combinatorial Mathematics IX, Lecture Notes in Mathematics, Vol. 952, Springer-Verlag, Berlin (1982), pp. 239–251.Google Scholar
  5. [F]
    Foulds, L. R., Graph theory applications,Universitext, Springer-Verlag, New York (1992).Google Scholar
  6. [FR1]
    Foulds, L. R., and R. W. Robinson, A strategy for solving the plant layout problem,Operational Research Quaterly 27 (1976), 845–855.Google Scholar
  7. [FR2]
    Foulds, L. R., and R. W. Robinson, Graph theoretic heuristics for the plant layout problem,International Journal of Production Research 16 (1978), 27–37.Google Scholar
  8. [GJ]
    Garey, M. R., and D. S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco (1979).Google Scholar
  9. [GT]
    Goldschmidt, O., and A. Takvorian, An Efficient Graph Planarization Two-Phase Heuristic, Technical Report ORP91-01, Dept. of Mech. Engr., University of Texas, Austin (1992).Google Scholar
  10. [GJR]
    Grötschel, M., M. Jünger, and G. Reinelt, A cutting plane algorithm for the linear ordering problem,Operations Research 32 (1984), 1195–1220.Google Scholar
  11. [H1]
    Himsolt, M., Konzeption und Implementierung von Grapheneditoren, Dissertation, Universität Passau (1993).Google Scholar
  12. [H2]
    Himsolt, M., Personal communication (1993).Google Scholar
  13. [HT]
    Hopcroft, J., and R. E. Tarjan, Efficient planarity testing,Journal of the Association for Computing Machinery 21 (1974), 549–568.Google Scholar
  14. [JTS]
    Jayakumar, R., K. Thulasiraman, and M. N. S. Swamy, O(n 2) algorithms for graph planarization,IEEE Transactions on Computer-aided Design 8 (1989), 257–267.Google Scholar
  15. [JRT]
    Jünger, M., G. Reinelt, and S. Thienel, Provably Good Solutions for the Traveling Salesman Problem, Rep. No. 92.114, Angewandte Mathematik und Informatik, Universität zu Köln (1992).Google Scholar
  16. [K]
    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
  17. [L]
    Leung, J., A new graph-theoretic heuristic for facility layout,Management Science 38 (1992), 594–605.Google Scholar
  18. [LG]
    Liu, P. C., and R. C. Geldmacher, On the deletion of nonplanar edges of a graph,Proc. 10th. S.-E. Conf. on Combinatorics, Graph Theory, and Computing, Boca Raton, FL (1977), pp. 727–738.Google Scholar
  19. [Ma]
    Martin, A., Personal communication (1993).Google Scholar
  20. [Mu]
    Mutzel, P., A Fast Linear Time Embedding Algorithm Based on the Hopcroft-Tarjan Planarity Test, Report No. 92.107, Angewandte Mathematik und Informatik, Universität zu Köln (1992).Google Scholar
  21. [PR]
    Padberg, M. W., and G. Rinaldi, A branch and cut algorithm for the resolution of large-scale symmetric traveling salesman problems,SIAM Review 33 (1991), 60–100.Google Scholar
  22. [P]
    Pulleyblank, W. R., Polyhedral combinatorics, in G. L. Nemhauser, A. H. G. Rinnoy Kan, and M. J. Todd (eds.),Handbook on Operations Research and Management Sciences: Networks, North-Holland, Amsterdam (1989).Google Scholar
  23. [TBB]
    Tamassia, R., G. Di Battista, and C. Batini, Automatic graph drawing and readability of diagrams,IEEE Transactions on Systems, Man, and Cybernetics 18 (1988), 61–79.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1996

Authors and Affiliations

  • M. Jünger
    • 1
  • P. Mutzel
    • 2
  1. 1.Institut für InformatikUniversität zu KölnKölnGermany
  2. 2.Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations