Skip to main content

Colouring AT-Free Graphs

  • Conference paper
Algorithms – ESA 2012 (ESA 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7501))

Included in the following conference series:

  • 2474 Accesses

Abstract

A vertex colouring assigns to each vertex of a graph a colour such that adjacent vertices have different colours. The algorithmic complexity of the Colouring problem, asking for the smallest number of colours needed to vertex-colour a given graph, is known for a large number of graph classes. Notably it is NP-complete in general, but polynomial time solvable for perfect graphs. A triple of vertices of a graph is called an asteroidal triple if between any two of the vertices there is a path avoiding all neighbours of the third one. Asteroidal triple-free graphs form a graph class with a lot of interesting structural and algorithmic properties. Broersma et al. (ICALP 1997) asked to find out the algorithmic complexity of Colouring on AT-free graphs. Even the algorithmic complexity of the k-Colouring problem, which asks whether a graph can be coloured with at most a fixed number k of colours, remained unknown for AT-free graphs. First progress was made recently by Stacho who presented an O(n 4) time algorithm for 3-colouring AT-free graphs (ISAAC 2010). In this paper we show that k-Colouring on AT-free graphs is in XP, i.e. polynomial time solvable for any fixed k. Even more, we present an algorithm using dynamic programming on an asteroidal decomposition which, for any fixed integers k and a, solves k-Colouring on any input graph G in time \(\mathcal{O}(f(a,k) \cdot n^{g(a,k)})\), where a denotes the asteroidal number of G, and f(a,k) and g(a,k) are functions that do not depend on n. Hence for any fixed integer k, there is a polynomial time algorithm solving k-Colouring on graphs of bounded asteroidal number. The algorithm runs in time \(\mathcal{O}(n^{8k+2})\) on AT-free graphs.

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

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. Brandstädt, A., Bang Le, V., Spinrad, J.: Graph classes: a survey. SIAM (1999)

    Google Scholar 

  2. Broersma, H.-J., Kloks, T., Kratsch, D., Müller, H.: Independent sets in asteroidal triple-free graphs. SIAM Journal on Discrete Mathematics 12, 276–287 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Broersma, H.-J., Kloks, T., Kratsch, D., Müller, H.: A generalization of AT-free graphs and a generic algorithm for solving triangulation problems. Algorithmica 32, 594–610 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Corneil, D.G., Olariu, S., Stewart, L.: Asteroidal triple-free graphs. SIAM Journal on Discrete Mathematics 10, 399–430 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Corneil, D.G., Olariu, S., Stewart, L.: Linear time algorithms for dominating pairs in asteroidal triple-free graphs. SIAM Journal on Computing 28, 1284–1297 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Couturier, J.-F., Golovach, P.A., Kratsch, D., Paulusma, D.: List Coloring in the Absence of a Linear Forest. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 119–130. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  7. Garey, M.R., Johnson, D.S.: Computers and Intractability: A guide to the Theory of NP-completeness. Freeman, New York (1979)

    MATH  Google Scholar 

  8. Grötschel, M., Lovász, L., Schrijver, A.: Polynomial algorithms for perfect graphs. Annals on Discrete Mathematics 21, 325–356 (1984)

    Google Scholar 

  9. Golovach, P.: Private communication

    Google Scholar 

  10. Johnson, D.S.: The NP-complete column: an ongoing guide. Journal of Algorithms 6, 434–451 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kloks, T., Kratsch, D., Müller, H.: Asteroidal Sets in Graphs. In: Möhring, R.H. (ed.) WG 1997. LNCS, vol. 1335, pp. 229–241. Springer, Heidelberg (1997)

    Google Scholar 

  12. Kloks, T., Kratsch, D., Müller, H.: Approximating the bandwidth for asteroidal triple-free graphs. Journal of Algorithms 32, 41–57 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kloks, T., Kratsch, D., Müller, H.: On the structure of graphs with bounded asteroidal number. Graphs and Combinatorics 17, 295–306 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kratsch, D.: Domination and total domination on asteroidal triple-free graphs. Discrete Applied Mathematics 99, 111–123 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kratsch, D., Müller, H., Todinca, I.: Feedback vertex set on AT-free graphs. Discrete Applied Mathematics 156, 1936–1947 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lekkerkerker, C.G., Boland, J.C.: Representation of a finite graph by a set of intervals on the real line. Fundamenta Mathematicae 51, 45–64 (1962)

    MathSciNet  MATH  Google Scholar 

  17. Stacho, J.: 3-Colouring AT-Free Graphs in Polynomial Time. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part II. LNCS, vol. 6507, pp. 144–155. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  18. Walter, J.R.: Representations of chordal graphs as subtrees of a tree. Journal of Graph Theory 2, 265–267 (1978)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kratsch, D., Müller, H. (2012). Colouring AT-Free Graphs. In: Epstein, L., Ferragina, P. (eds) Algorithms – ESA 2012. ESA 2012. Lecture Notes in Computer Science, vol 7501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33090-2_61

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-33090-2_61

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33089-6

  • Online ISBN: 978-3-642-33090-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics