Advertisement

On Strong Tree-Breadth

  • Arne LeitertEmail author
  • Feodor F. Dragan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10043)

Abstract

In this paper, we introduce and investigate a new notion of strong tree-breadth. We say that a graph G has strong tree-breadth \(\rho \) if there is a tree-decomposition T for G such that each bag B of T is equal to the complete \(\rho \)-neighbourhood of some vertex v in G, i. e., \(B = N_G^\rho [v]\). We show that
  • it is NP-complete to determine if a given graph has strong tree-breadth \(\rho \), even for \(\rho = 1\);

  • if a graph G has strong tree-breadth \(\rho \), then we can find a tree-decomposition for G with tree-breadth \(\rho \) in \(\mathcal {O}(n^2m)\) time;

  • with some additional restrictions, a tree-decomposition with strong breadth \(\rho \) can be found in polynomial time;

  • some graph classes including distance-hereditary graphs have strong tree-breadth 1.

References

  1. 1.
    Abu-Ata, M., Dragan, F.F.: Metric tree-like structures in real-life networks: an empirical study. Networks 67(1), 49–68 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bandelt, H.-J., Mulder, H.M.: Distance-hereditary graphs. J. Comb. Theory Ser. B 41, 182–208 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brandstädt, A., Chepoi, V.D., Dragan, F.F.: The algorithmic use of hypertree structure and maximum neighborhood orderings. Discret. Appl. Math. 82, 43–77 (1998)CrossRefzbMATHGoogle Scholar
  4. 4.
    Brandstädt, A., Dragan, F.F., Chepoi, V.D., Voloshin, V.: Dually chordal graphs. SIAM J. Discret. Math. 11(3), 437–455 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brandstädt, A., Fičur, P., Leitert, A., Milanič, M.: Polynomial-time algorithms for weighted efficient domination problems in AT-free graphs and dually chordal graphs. Inf. Process. Lett. 115(2), 256–262 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brandstädt, A., Leitert, A., Rautenbach, D.: Efficient dominating and edge dominating sets for graphs and hypergraphs. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 267–277. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  7. 7.
    Damiand, G., Habib, M., Paul, C.: A simple paradigm for graph recognition: application to cographs and distance hereditary graphs. Theoret. Comput. Sci. 263(1–2), 99–111 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dourisboure, Y., Gavoille, C.: Tree-decompositions with bags of small diameter. Discret. Math. 307(16), 2008–2029 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dragan, F.F., Köhler, E.: An approximation algorithm for the tree t-spanner problem on unweighted graphs via generalized chordal graphs. Algorithmica 69, 884–905 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dragan, F.F., Köhler, E., Leitert, A.: Line-distortion, bandwidth and path-length of a graph. Algorithmica (in print)Google Scholar
  11. 11.
    Dragan, F.F., Lomonosov, I.: On compact and efficient routing in certain graph classes. Discret. Appl. Math. 155, 1458–1470 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dragan, F.F., Matamala, M.: Navigating in a graph by aid of its spanning tree. SIAM J. Discret. Math. 25(1), 306–332 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ducoffe, G., Legay, S., Nisse, N.: On computing tree and path decompositions with metric constraints on the bags. CoRR abs/1601.01958 (2016)Google Scholar
  14. 14.
    Halin, R.: S-functions for graphs. J. Geom. 8(1–2), 171–186 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Robertson, N., Seymour, P.D.: Graph minors. I. Excluding a forest. J. Comb. Theory Ser. B 35(1), 39–61 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Robertson, N., Seymour, P.D.: Graph minors. III. Planar tree-width. J. Comb. Theory Ser. B 36(1), 49–64 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing (STOC 1978), pp. 216–226 (1978)Google Scholar
  18. 18.
    Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13(3), 566–579 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceKent State UniversityKentUSA

Personalised recommendations