Maximum Weighted Induced Bipartite Subgraphs and Acyclic Subgraphs of Planar Cubic Graphs

  • Mourad Baïou
  • Francisco Barahona
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)


We study the maximum node-weighted induced bipartite subgraph problem in planar graphs with maximum degree three. We show that this is polynomially solvable. It was shown in [6] that it is NP-complete if the maximum degree is four. We extend these ideas to the problem of balancing signed graphs.

We also consider maximum weighted induced acyclic subgraphs of planar directed graphs. If the maximum degree is three, it is easily shown that this is polynomially solvable. We show that for planar graphs with maximum degree four the same problem is NP-complete.


Maximum induced bipartite subgraph balancing signed graphs maximum induced acyclic subgraph polynomial algorithm NP-completeness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barahona, F.: Planar multicommodity flows, max cut and the Chinese Postman Problem. In: Polyhedral Combinatorics. DIMACS Series on Discrete Mathematics and Theoretical Computer Science No. 1, pp. 189–202. DIMACS, NJ (1990)Google Scholar
  2. 2.
    Barahona, F., Mahjoub, A.: Facets of the balanced (acyclic) induced subgraph polytope. Mathematical Programming 45, 21–33 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Barahona, F., Mahjoub, A.: Compositions of graphs and polyhedra I: Balanced induced subgraphs and acyclic subgraphs. SIAM Journal on Discrete Mathematics 7, 344–358 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Barahona, F., Maynard, R., Rammal, R., Uhry, J.: Morphology of ground states of two-dimensional frustration model. Journal of Physics A: Mathematical and General 15, 673 (1982)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Chen, R.-W., Kajitani, Y., Chan, S.-P.: A graph-theoretic via minimization algorithm for two-layer printed circuit boards. IEEE Transactions on Circuits and Systems 30, 284–299 (1983)CrossRefzbMATHGoogle Scholar
  6. 6.
    Choi, H.-A., Nakajima, K., Rim, C.S.: Graph bipartization and via minimization. SIAM Journal on Discrete Mathematics 2, 38–47 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Edmonds, J., Johnson, E.L.: Matching, euler tours and the chinese postman. Mathematical Programming 5, 88–124 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Even, G., Naor, J.S., Schieber, B., Sudan, M.: Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica 20, 151–174 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Festa, P., Pardalos, P.M., Resende, M.G.: Feedback set problems. In: Handbook of Combinatorial Optimization, pp. 209–258. Springer (1999)Google Scholar
  10. 10.
    Fouilhoux, P., Mahjoub, A.: Solving VLSI design and DNA sequencing problems using bipartization of graphs. Computational Optimization and Applications 51, 749–781 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Funke, M., Reinelt, G.: A polyhedral approach to the feedback vertex set problem. In: Integer Programming and Combinatorial Optimization, pp. 445–459. Springer (1996)Google Scholar
  12. 12.
    Gabow, H.N.: An efficient implementation of edmonds’ algorithm for maximum matching on graphs. Journal of the ACM (JACM) 23, 221–234 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Gabow, H.N.: Centroids, representations, and submodular flows. Journal of Algorithms 18, 586–628 (1995)Google Scholar
  14. 14.
    Gardarin, G., Spaccapietra, S.: Integrity of data bases: A general lockout algorithm with deadlock avoidance. In: IFIP Working Conference on Modelling in Data Base Management Systems, pp. 395–412 (1976)Google Scholar
  15. 15.
    Garey, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of NP-completeness (1979)Google Scholar
  16. 16.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Approximate max-flow min-(multi) cut theorems and their applications. SIAM Journal on Computing 25, 235–251 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Goemans, M.X., Williamson, D.P.: Primal-dual approximation algorithms for feedback problems in planar graphs. Combinatorica 18, 37–59 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Hadlock, F.: Finding a maximum cut of a planar graph in polynomial time. SIAM Journal on Computing 4, 221–225 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J., Bohlinger, J. (eds.) Complexity of Computer Computations. The IBM Research Symposia Series, pp. 85–103. Springer US (1972)Google Scholar
  20. 20.
    Leiserson, C.E., Saxe, J.B.: Retiming synchronous circuitry. Algorithmica 6, 5–35 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Lichtenstein, D.: Planar formulae and their uses. SIAM Journal on Computing 11, 329–343 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM Journal on Applied Mathematics 36, 177–189 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Lucchesi, C., Younger, D.: A minimax theorem for directed graphs. J. London Math. Soc. 17(2), 369–374 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Lucchesi, C.L.: A minimax equality for directed graphs, PhD thesis, Thesis (Ph. D.)–University of Waterloo (1976)Google Scholar
  25. 25.
    Pinter, R.Y.: Optimal layer assignment for interconnect. Journal of VLSI and Computer Systems 1, 123–137 (1984)zbMATHGoogle Scholar
  26. 26.
    Seymour, P.D.: On odd cuts and plane multicommodity flows. Proceedings of the London Mathematical Society 3, 178–192 (1981)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Silberschatz, A., Peterson, J.L., Galvin, P.B.: Operating system concepts. Addison-Wesley Longman Publishing Co., Inc. (1991)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Mourad Baïou
    • 1
  • Francisco Barahona
    • 2
  1. 1.CNRS and Université Clermont IIAubière CedexFrance
  2. 2.IBM T. J. Watson research CenterYorktown HeightsUSA

Personalised recommendations