Advertisement

On some generalizations of outerplanar graphs: Results and open problems

  • Maciej M. Sysło
Outerplanar Graphs And Graph Isomorphism
Part of the Lecture Notes in Computer Science book series (LNCS, volume 246)

Abstract

Outerplanar graphs have been recently generalized in many directions. Almost all generalizations have been introduced to parameterize the family of planar graphs so that in consequence some of the decision problems which are NP-complete for planar graphs and easy (or trivial) for outerplanar graphs can be solved in polynomial time for every fixed value of a parameter. In this paper, we survey some families of graphs which are known as generalizations of outerplanar graphs: tree-structured graphs (e.g., series-parallel, k-terminal and Halin graphs), W-outerplanar and k-outerplanar. We discuss also some problems which are formulated by slightly modified versions of the statements that characterize outerplanar graphs: the largest face problem and the (independent) face covers in plane graphs. Almost every (sub)section of the paper contains an open problem, a good starting point for further research. Our description of results is rather informal, the reader interested in details is referred to the original works.

Keywords

Planar Graph Hamiltonian Cycle Vertex Cover Decomposition Tree Dual Graph 
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.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Arnborg, Efficient algorithms for combinatorial problems on graphs with bounded decomposability — a survey, BIT 25 (1985), 2–23.Google Scholar
  2. [2]
    B.S. Baker, Approximation algorithms for NP-complete problems on planar graphs, Proc. 24th Ann. IEEE Symp. on FOCS, 1983, pp. 265–273.Google Scholar
  3. [3]
    M.W. Bern, E.L. Lawler, A.L. Wong, Why certain subgraph computations require only linear time, Proc. 26th Annal IEEE FOCS Symposium, 1985, 117–125.Google Scholar
  4. [4]
    F.R.K. Chung, F.T. Leighton, A.L. Rosenberg, Embedding graphs in books: a layout problem with applications to VLSI design, SIAM J. Alg. Discrete Meth., to appear.Google Scholar
  5. [5]
    G. Cornuéjols, D. Naddef, W. Pulleyblank, Halin graphs and the travelling salesman problem, Math. Programming 26 (1983), 287–294.Google Scholar
  6. [6]
    N. Deo, G.M. Prabhu, M.S. Krishnamoorthy, Algorithms for generating fundamental cycles in a graph, ACM Trans. Math. Software 8 (1982), 26–42.Google Scholar
  7. [7]
    M. Fellows, F. Hickling, M.M. Sysło, A topological parameterization of hard graph problem (Extended Abstract), manuscript, January 1986.Google Scholar
  8. [8]
    M. Fellows, personal communication, July 1986.Google Scholar
  9. [9]
    F. Harary, Graph Theory, Addison-Wesley, Reading, Mass., 1969.Google Scholar
  10. [10]
    J.D. Horton, A polynomial-time algorithm to find the shortest cycle basis of a graph, manuscript, March 1985.Google Scholar
  11. [11]
    D.S. Johnson, The NP-completeness column: an ongoing guide (16), J. Algorithms 6 (1985), 434–451.CrossRefGoogle Scholar
  12. [12]
    V.P. Nekrasov, On planar graphs having an external access (in Russian), in: Combinatorial Properties of Convex Sets and Graphs, AN USSR, Sverdlovsk, 1983, pp. 34–44.Google Scholar
  13. [13]
    L. Oubina, R. Zucchello, A generalization of outerplanar graphs, Discrete Math. 51 (1984), 243–249.CrossRefGoogle Scholar
  14. [14]
    N. Robertson, P.D. Seymour, Disjoint paths — a survey, SIAM J. Alg. Discrete Meth. 6 (1985), 300–305.Google Scholar
  15. [15]
    N. Robertson, P.D. Seymour, Graph minors — a survey, manuscript.Google Scholar
  16. [16]
    D. Seese, Tree-partite graphs and the complexity of algorithms, P-MATH-08/86, AW der DDR, Inst. für Mathematik, Berlin 1986.Google Scholar
  17. [17]
    S. Shinoda, Y. Kajitani, K. Onaga, W. Mayeda, Various characterizations of series-parallel graphs, Proc. 1979 ISCHS, pp. 100–103.Google Scholar
  18. [18]
    M. Skowrońska, The pancyclicity of Halin graphs and their exterior contractions, Am. Discrete Math. 27 (1985), 179–194.Google Scholar
  19. [19]
    M.M. Sysło, Outerplanar graphs: characterizations, testing, coding, and counting, Bull. Acad. Polon. Sci., Ser., Sci. Math. Astronom. Phys. 26 (1978), 675–684.Google Scholar
  20. [20]
    M.M. Sysło, Characterizations of outerplanar graphs, Discrete Math. 26 (1979), 47–53.CrossRefGoogle Scholar
  21. [21]
    M.M. Sysło, A. Proskurowski, On Halin graphs, in: M. Borowiecki, J.W. Kennedy, M.M. Sysło (eds.), Graph Theory — Łagów 1981, Springer-Verlag, Berlin, 1983, pp. 248–256.Google Scholar
  22. [22]
    M.M. Sysło, NP-complete problems on some tree-structured graphs: a review, in: M. Nagl, J. Perl (eds.), Proceedings WG'83, Trauner Verlag, Linz, 1984, pp. 342–353.Google Scholar
  23. [23]
    M.M. Sysło, On two problems related to the traveling salesman problem on Halin graphs, in: G. Hammer, D. Pallaschke (eds.), Selected Topics in Operations Research and Mathematical Economics, Springer-Verlag, Berlin 1984, pp. 325–335.Google Scholar
  24. [24]
    M.M. Sysło, Series-parallel graphs and depth-first search trees, IEEE Trans. on Circuits and Systems CAS-31 (1984), 1029–1033.CrossRefGoogle Scholar
  25. [25]
    K. Takamizawa, T. Nishizeki, N. Saito, Linear-time computability of combinatorial problems on series-parallel graphs, J. ACM 29 (1982), 623–641.CrossRefGoogle Scholar
  26. [26]
    C. Thomassen, Planarity and duality of finite and infinite graphs, J. Combinatorial Theory B-29 (1980), 244–271.Google Scholar
  27. [27]
    W.T. Tutte, Separation of vertices by a circuit, Discrete Math. 12 (1975), 173–184.Google Scholar
  28. [28]
    T.V. Wimer, S.T. Hedetniemi, R. Laskar, A methodology for constructing linear graph algorithms, Congressus Numerantium 50 (1985), 43–60.Google Scholar
  29. [29]
    T.V. Wimer, S.T. Hedetniemi, K-terminal recursive families of graphs, TR 86-MAY-7, Dept. Comput. Science, Clemson University, 1986.Google Scholar
  30. [30]
    F.F. Yao, Graph 2-isomorphism is NP-complete, Information Proc. Lett. 9 (1979), 68–72.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Maciej M. Sysło
    • 1
  1. 1.Inst. Comput. Sci.University of WrocławWrocławPoland

Personalised recommendations