Broadcast Domination on Block Graphs in Linear Time

  • Pinar Heggernes
  • Sigve H. Sæther
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7353)

Abstract

A broadcast domination on a graph assigns an integer value f(u) ≥ 0 to each vertex u, such that every vertex u with f(u) = 0 is within distance f(v) from a vertex v with f(v) > 0. The Broadcast Domination problem seeks to compute a broadcast domination where the sum of the assigned values is minimized. We show that Broadcast Domination can be solved in linear time on block graphs. For general graphs the best known algorithm runs in time \(\mathcal{O}(n^6)\). For trees and interval graphs, linear-time algorithms are known. As block graphs form a superclass of trees, our result extends the classes of graphs on which this problem is solvable in linear time.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bandelt, H.J., Mulder, H.M.: Distance-hereditary graphs. J. Comb. Theor., Ser. B 41, 182–208 (1986)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Chang, R.Y., Peng, S.L.: A linear-time algorithm for broadcast domination problem on interval graphs. In: Proceedings of the 27th Workshop on Combinatorial Mathematics and Computation Theory, pp. 184–188. Providence University, Taichung (2010)Google Scholar
  3. 3.
    Cockayne, E.J., Herke, S., Mynhardt, C.M.: Broadcasts and domination in trees. Disc. Math. 311, 1235–1246 (2011)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Dabney, J., Dean, B.C., Hedetniemi, S.T.: A linear-time algorithm for broadcast domination in a tree. Networks 53(2), 160–169 (2009)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Dunbar, J.E., Erwin, D.J., Haynes, T.W., Hedetniemi, S.M., Hedetniemi, S.T.: Broadcasts in graphs. Disc. Appl. Math. 154(1), 59–75 (2006)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Erwin, D.J.: Dominating broadcasts in graphs. Bull. Inst. Comb. Appl. 42, 89–105 (2004)MathSciNetMATHGoogle Scholar
  7. 7.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker (1998)Google Scholar
  8. 8.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J. (eds.): Domination in Graphs: Advanced Topics. Marcel Dekker (1998)Google Scholar
  9. 9.
    Hedetniemi, S.T., Laskar, R.C. (eds.): Topics on domination. North Holland (1990)Google Scholar
  10. 10.
    Heggernes, P., Lokshtanov, D.: Optimal broadcast domination of arbitrary graphs in polynomial time. Disc. Math. 306(24), 3267–3280 (2006)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Herke, S., Mynhardt, C.M.: Radial trees. Disc. Math. 309, 5950–5962 (2009)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Howorka, E.: On metric properties of certain clique graphs. J. Comb. Theor., Ser. B 27(1), 67–74 (1979)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Pinar Heggernes
    • 1
  • Sigve H. Sæther
    • 1
  1. 1.Department of InformaticsUniversity of BergenNorway

Personalised recommendations