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Forbidden Directed Minors, Directed Path-Width and Directed Tree-Width of Tree-Like Digraphs

  • Frank Gurski
  • Carolin Rehs
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11376)

Abstract

There have been many attempts to find directed graph classes with bounded directed path-width and bounded directed tree-width. Right now, the only known directed tree-width-/path-width-bounded graphs are cycle-free graphs with directed path-width and directed tree-width 0. In this paper, we introduce directed versions of cactus trees and pseudotrees and -forests and characterize them by at most three forbidden directed graph minors. Furthermore, we show that directed cactus trees and forests have a directed tree-width of at most 1 and directed pseudotrees and -forests even have a directed path-width of at most 1.

Keywords

Directed cactus trees Directed pseudoforests Directed graph minors Directed path-width Directed tree-width 

References

  1. 1.
    Bang-Jensen, J., Gutin, G. (eds.): Classes of Directed Graphs. SMM. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-71840-8CrossRefzbMATHGoogle Scholar
  2. 2.
    Barát, J.: Directed pathwidth and monotonicity in digraph searching. Graphs Comb. 22, 161–172 (2006)CrossRefGoogle Scholar
  3. 3.
    El-Mallah, E., Colbourn, C.J.: The complexity of some edge deletion problems. IEEE Trans. Circuits Syst. 35(3), 354–362 (1988)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ganian, R., et al.: Are there any good digraph width measures? J. Comb. Theory Ser. B 116, 250–286 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gurski, F., Rehs, C.: Computing directed path-width and directed tree-width of recursively defined digraphs. ACM Computing Research Repository, abs/1806.04457, p. 16 (2018)Google Scholar
  6. 6.
    Gurski, F., Rehs, C.: Directed path-width and directed tree-width of directed co-graphs. In: Wang, L., Zhu, D. (eds.) COCOON 2018. LNCS, vol. 10976, pp. 255–267. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-94776-1_22CrossRefzbMATHGoogle Scholar
  7. 7.
    Harary, F., Uhlenbeck, G.E.: On the number of husimi trees: I. Proc. Nat. Acad. Sci. 39(4), 315–322 (1953)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. J. Comb. Theory Ser. B 82, 138–155 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kintali, S., Zhang, Q.: Forbidden directed minors and directed pathwidth. Reseach report (2015)Google Scholar
  10. 10.
    Kintali, S., Zhang, Q.: Forbidden directed minors and Kelly-width. Theor. Comput. Sci. 662, 40–47 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Paten, B., et al.: Cactus graphs for genome comparisons. J. Comput. Biol. 18(3), 469–481 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Paten, B., Earl, D., Nguyen, N., Diekhans, M., Zerbino, D., Haussler, D.: Cactus: algorithms for genome multiple sequence alignment. Genome Res. 21(9), 1512–11528 (2011)CrossRefGoogle Scholar
  13. 13.
    Robertson, N., Seymour, P.D.: Graph minors II. Algorithmic aspects of tree width. J. Algorithms 7, 309–322 (1986)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Scheffler, P.: Die baumweite von graphen als mass für die kompliziertheit algorithmischer probleme. Ph.D. thesis, Akademie der Wissenschaften in der DDR, Berlin (1989)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Computer Science, Algorithmics for Hard Problems GroupHeinrich-Heine-University DüsseldorfDüsseldorfGermany

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