Skip to main content
Springer Nature Link
Log in

LexBFS-orderings and powers of graphs

  • Conference paper
  • First Online:
Graph-Theoretic Concepts in Computer Science (WG 1996)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1197))

Included in the following conference series:

Abstract

For an undirected graph G the k-th power G k of G is the graph with the same vertex set as G where two vertices are adjacent iff their distance is at most k in G. In this paper we consider LexBFS-orderings of chordal, distance-hereditaxy and HHD-free graphs (the graphs where each cycle of length at least five has two chords) with respect to their powers. We show that any LexBFS-ordering of a chordal graph is a common perfect elimination ordering of all odd powers of this graph, and any LexBFS-ordering of a distance-hereditary graph is a common perfect elimination ordering of all its even powers. It is wellknown that any LexBFS-ordering of a HHD-free graph is a so-called semi-simplicial ordering. We show, that any LexBFS-ordering of a HHD-free graph is a common semi-simplicial ordering of all its odd powers. Moreover we characterize those chordal, distance-hereditary and HHD-free graphs by forbidden isometric subgraphs for which any LexBFS-ordering of the graph is a common perfect elimination ordering of all its nontrivial powers. As an application we get a linear time approximation of the diameter for weak bipolarizable graphs, a subclass of HHD-free graphs containing all chordal graphs, and an algorithm which computes the diameter and a diametral pair of vertices of a distance-hereditary graph in linear time.

First author supported by DAAD, second author supported by DFG.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. H.J. Bandelt, A. Henkmann and F. Nicolai, Powers of distance-hereditary graphs, Discr. Math. 145 (1995), 37–60.

    Google Scholar 

  2. H.-J. Bandelt and H.M. Mulder, Distance-hereditary graphs, Journal of Combin. Theory (B) 41 (1986), 182–208.

    Google Scholar 

  3. A. Brandstädt, V.D. Chepoi and F.F. Dragan, Perfect elimination orderings of chordal powers of graphs, Technical Report Gerhard-Mercator-Universität — Gesamthochschule Duisburg SM-DU-252, 1994 (to appear in Discr. Math.).

    Google Scholar 

  4. A. Brandstädt, F.F. Dragan, V.D. Chepoi and V.I. Voloshin, Dually chordal graphs, Proc. of WG'93, Springer, Lecture Notes in Computer Science 790 (1994), 237–251.

    Google Scholar 

  5. A. Brandstädt, F.F. Dragan and V.B. Le, Induced cycles and odd powers of graphs, Technical Report Universität Rostock CS-09-95, 1995.

    Google Scholar 

  6. A. Brandstädt, F.F. Dragan and F. Nicolai, LexBFS-orderings and powers of chordal graphs, Technical Report Gerhard-Mercator-Universität — Gesamthochschule Duisburg SM-DU-287, 1995 (to appear in Discr. Math.).

    Google Scholar 

  7. R. Chandrasekaran and A. Doughety, Location on tree networks: p-center and q-dispersion problems, Math. Oper. Res. 6 (1981), No. 1, 50–57.

    Google Scholar 

  8. G.J. Chang and G.L. Nemhauser, The k-domination and k-stability problems on sun-free chordal graphs, SIAM J. Algebraic and Discrete Methods, 5 (1984), 332–345.

    Google Scholar 

  9. G. Chartrand and D.C. Kay, A characterization of certain ptolemaic graphs, Canad. Journal Math. 17 (1965), 342–346.

    Google Scholar 

  10. V. Chvatal, Perfectly orderable graphs, Annals of Discrete Math. 21 (1984), 63–65.

    Google Scholar 

  11. A. D'Atri and M. Moscarini, Distance-hereditary graphs, Steiner trees and connected domination, SIAM J. Computing 17 (1988), 521–538.

    Google Scholar 

  12. F.F. Dragan, Dominating cliques in distance-hereditary graphs, Proceedings of SWAT'94, Springer, Lecture Notes in Computer Science 824, 370–381.

    Google Scholar 

  13. F.F. Dragan and F. Nicolai, LexBFS-orderings of distance-hereditary graphs, Technical Report Gerhard-Mercator-Universität — Gesamthochschule Duisburg SM-DU-303, 1995.

    Google Scholar 

  14. F.F. Dragan and F. Nicolai, LexBFS-orderings and powers of HHD-free graphs, Technical Report Gerhard-Mercator-Universität — Gesamthochschule Duisburg SM-DU-322, 1996.

    Google Scholar 

  15. F.F. Dragan, F. Nicolai and A. Brandstädt, Convexity and HHD-free graphs, Technical Report Gerhard-Mercator-Universität — Gesamthochschule Duisburg SM-DU-290, 1995.

    Google Scholar 

  16. F.F. Dragan, F. Nicolai and A. Brandstädt, Powers of HHD-free graphs, Technical Report Gerhard-Mercator-Universität — Gesamthochschule Duisburg SM-DU-315, 1995.

    Google Scholar 

  17. F.F. Dragan, C.F. Prisacaru and V.D. Chepoi, Location problems in graphs and the Helly property (in Russian), Discrete Mathematics, Moscow, 4 (1992), 67–73.

    Google Scholar 

  18. P. Duchet, Classical perfect graphs, Annals of Discr. Math. 21 (1984), 67–96.

    Google Scholar 

  19. M. Farber and R.E. Jamison, Convexity in graphs and hypergraphs, SIAM Journal Alg. Discrete Meth. 7, 3 (1986), 433–444.

    Google Scholar 

  20. M. Gionfriddo, A short survey on some generalized colourings of graphs, Ars Comb. 30 (1986), 275–284.

    Google Scholar 

  21. M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York 1980.

    Google Scholar 

  22. P.L. Hammer and F. Maffray, Completely separable graphs, Discr. Appl. Math. 27 (1990), 85–99.

    Google Scholar 

  23. E. Howorka, A characterization of distance-hereditary graphs, Quart. J. Math. Oxford Ser. 2, 28 (1977), 417–420.

    Google Scholar 

  24. E. Howorka, A characterization of ptolemaic graphs, Journal of Graph Theory 5 (1981), 323–331.

    Google Scholar 

  25. B. Jamison and S. Olariu, On the semi-perfect elimination, Advances in Applied Math. 9 (1988), 364–376.

    Google Scholar 

  26. T.R. Jensen and B. Toft, Graph coloring problems, Wiley 1995.

    Google Scholar 

  27. T. Kloks, D. Kratsch and H. Müller, Approximating the bandwidth for AT-free graphs, Proceedings of European Symposium on Algorithms ESA'95, Springer, Lecture Notes in Computer Science 979 (1995), 434–447.

    Google Scholar 

  28. R. Laskar and D.R. Shier, On powers and centers of chordal graphs, Discr. Appl. Math. 6 (1983), 139–147.

    Google Scholar 

  29. F. Nicolai, A hypertree characterization of distance-hereditary graphs, Technical Report Gerhard-Mercator-Universität — Gesamthochschule Duisburg SM-DU-255 1994.

    Google Scholar 

  30. F. Nicolai, Hamiltonian problems on distance-hereditary graphs, Technical Report Gerhard-Mercator-Universität — Gesamthochschule Duisburg SM-DU-264, 1994.

    Google Scholar 

  31. S. Olariu, Weak bipolarizable graphs, Discr. Math. 74 (1989), 159–171.

    Google Scholar 

  32. D. Rose, R.E. Tarjan and G. Lueker, Algorithmic aspects on vertex elimination on graphs, SIAM J. Computing 5 (1976), 266–283.

    Google Scholar 

  33. R.E. Tarjan and M. Yannakakis, Simple linear time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs, SIAM J. Computing 13, 3 (1984), 566–579.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Cybernetics, Moldova State University, A. Mateevici str. 60, 277009, ChiŞinĂu, Moldova

    Feodor F. Dragan

  2. FB Mathematik, FG Informatik, Gerhard-Mercator-Universität -GH- Duisburg, D 47048, Duisburg, Germany

    Falk Nicolai

  3. FB Informatik, Lehrstuhl für Theoretische Informatik, Universität Rostock, D 18051, Rostock, Germany

    Andreas Brandstädt

Authors
  1. Feodor F. Dragan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Falk Nicolai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andreas Brandstädt
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Fabrizio d'Amore Paolo Giulio Franciosa Alberto Marchetti-Spaccamela

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Dragan, F.F., Nicolai, F., Brandstädt, A. (1997). LexBFS-orderings and powers of graphs. In: d'Amore, F., Franciosa, P.G., Marchetti-Spaccamela, A. (eds) Graph-Theoretic Concepts in Computer Science. WG 1996. Lecture Notes in Computer Science, vol 1197. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62559-3_15

Download citation

  • DOI: https://doi.org/10.1007/3-540-62559-3_15

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62559-9

  • Online ISBN: 978-3-540-68072-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics