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An efficient parallel strategy for the perfect domination problem on distance-hereditary graphs

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Abstract

A graph is distance-hereditary if the distance stays the same between any of two vertices in every connected induced subgraph containing both. Two well-known classes of graphs, trees and cographs, both belong to distance-hereditary graphs. In this paper, we first show that the perfect domination problem can be solved in sequential linear-time on distance-hereditary graphs. By sketching some regular property of the problem, we also show that it can be easily parallelized on distance-hereditary graphs.

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References

  1. Abrahamson K, Dadoun N, Kirkpatrick DG, Przytycka T (1989) A simple parallel tree contraction algorithm. J Alg 10:287–302

    MathSciNet  Google Scholar 

  2. Bandelt HJ, Mulder HM (1989) Distance-hereditary graphs. J Comb Theory Series B 41(1):182–208

    MathSciNet  Google Scholar 

  3. Berge C (1973) Graphs and hypergraphs. North-Holland, Amsterdam

    MATH  Google Scholar 

  4. Brandstädt A, Dragan FF (1998) A linear time algorithm for connected γ-domination and Steiner tree on distance-hereditary graphs. Networks 31:177–182

    Article  MathSciNet  Google Scholar 

  5. Chang GJ (1988) Labeling algorithms for domination problems in sun-free chordal graphs. Discr Appl Math 22(1):21–34

    Article  Google Scholar 

  6. Chang GJ, Nemhauser GL (1984) The k-domination and k-stability problems on sun-free chordal graphs. SIAM J Alg Discr Meth 5:332–345

    MathSciNet  Google Scholar 

  7. Chang MS, Hsieh SY, Chen GH (1997) Dynamic programming on distance-hereditary graphs. In: Proceedings of 7th international symposium on algorithms and computation (ISAAC’97), LNCS 1350, pp 344–353

  8. Cockayne EJ, Goodman S, Hedetniemi ST (1975) A linear algorithm for the domination number of a tree. Inf Proc Lett 4:41–44

    Article  Google Scholar 

  9. Courcelle B, Makowsky JA, Rotics U (2000) Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comp Syst 33:125–150

    Article  MathSciNet  Google Scholar 

  10. Dahlhaus E (1995) Efficient parallel recognition algorithms of cographs and distance-hereditary graphs. Dis Appl Math 57(1):29–44

    Article  MathSciNet  Google Scholar 

  11. Damiand G, Habib M, Paul C (2001) A simple paradigm for graph recognition: application to cographs and distance-hereditary graphs. Theory Comp Sci 263:99–111

    Article  MathSciNet  Google Scholar 

  12. D’atri A, Moscarini M (1988) Distance-hereditary graphs, steiner trees, and connected domination. SIAM J Comp 17(3):521–538

    Article  MathSciNet  Google Scholar 

  13. Dragan FF (1994) Dominating cliques in distance-hereditary graphs. In: Proceedings of 4th scandinavian workshop on algorithm theory (SWAT’94), LNCS 824, pp 370–381

  14. Golumbic MC (1980) Algorithmic graph theory and perfect graphs. Academic Press, New York

    MATH  Google Scholar 

  15. Golumbic MC, Rotics U (2000) On the clique-width of some perfect graph classes. Int J Found Comp Sci 11(3):423–443

    Article  MathSciNet  Google Scholar 

  16. Hammer PL, Maffray F (1990) Complete separable graphs. Discr Appl Math 27(1):85–99

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  18. Hsieh SY, Ho CW, Hsu T-s, Ko MT, Chen GH (1999) Efficient parallel algorithms on distance-hereditary graphs. Par Proc Lett 9(1):43–52

    Article  MathSciNet  Google Scholar 

  19. Hsieh SY, Ho CW, Hsu T-s, Ko MT, Chen GH (2002) Characterization of efficient parallel solvable problems on distance-hereditary graphs. SIAM J Discr Math 15(4):488–518

    Article  MathSciNet  Google Scholar 

  20. Hsieh SY, Ho CW, Hsu T-s, Ko MT, Chen GH (2000) A faster implementation of a parallel tree contraction scheme and its application on distance-hereditary graphs. J Algo 35:50–81

    Article  MathSciNet  Google Scholar 

  21. Hsieh SY (1999) Parallel decomposition of distance-hereditary graphs. In: Proceedings of the 4th international ACPC conference including special tracks on parallel numerics (ParNum’99) and parallel computing in image processing, video processing, and multimedia (ACPC’99), LNCS 1557, pp 417–426

  22. Hsieh SY (2002) An efficient parallel algorithm for the efficient domination problem on distance-hereditary graphs. IEEE Trans Par Distr Syst 13(9):985–993

    Article  MathSciNet  Google Scholar 

  23. Hsieh SY, Ho CW, Hsu T-s, Ko MT (2002) The Hamiltonian problem on distance-hereditary graphs, submitted. extended abstract. In: Proceedings of the eight annual international computing and combinatorics conference COCOON’02, LNCS 2387, pp 77–86, under the title “Efficient algorithms for the Hamiltonian problem on distance-hereditary graphs”

  24. Hung RW, Wu SC, Chang MS (2003) Hamiltonian cycle problem on distance-hereditary graphs. J Inf Sci Eng 19:827–838

    MathSciNet  Google Scholar 

  25. Müller H, Nicolai F (1993) Polynomial time algorithms for Hamiltonian problems on bipartite distance-hereditary graphs. Inf Proc Lett 46:225–230

    Article  Google Scholar 

  26. Nicolai F (1996) Hamiltonian problems on distance-hereditary graphs, Technique report SM-DU-264, Gerhard-Mercator-University Duisburg (corrected version)

  27. Ja’Ja’ J (1992) An introduction to parallel algorithms. Addison Wesley

  28. Karp RM, Ramachandran V (1990) Parallel algorithms for shared memory machines. Handbook of theoretical computer science, North-Holland, Amsterdan, pp 869–941

    Google Scholar 

  29. Laskar R, Pfaff J, Hedetniemi SM, Hedetniemi ST (1984) On the algorithmic complexity of a total domination. SIAM J Alg Discr Meth 5:420–425

    MathSciNet  Google Scholar 

  30. Laskar R, Walikan HB (1980) On domination related concepts in graph theory. In: Proceedings of combinatorics and graph theory, lecture notes in mathematics, vol 885, pp 308–320

  31. Slater PJ (1976) R-domination in graphs. J ACM 23:446–450

    Article  MathSciNet  Google Scholar 

  32. Weichsel PM (1994) Dominating sets in n-cubes. J Graph Theory 18(5):479–488

    MathSciNet  Google Scholar 

  33. Yeh HG, Chang GJ (1998) Weighted connected domination and Steiner trees in distance-hereditary graphs. Discrete Appl Math 87(1–3):245–253

    MathSciNet  Google Scholar 

  34. Yeh HG, Chang GJ (manuscript) Linear-time algorithms for bipartite distance-hereditary graphs

  35. Yeh HG, Chang GJ (1998) The path-partition problem in bipartite distance-hereditary graphs. Taiwaness J Math 2(3):353–360

    MathSciNet  Google Scholar 

  36. Yeh HG, Chang GJ (2001) Weighted connected k-domination and weighted k-dominating clique in distance-hereditary graphs. Theory Comp Sci 263:3–8

    Article  MathSciNet  Google Scholar 

  37. Yen CC, Lee RCT (1990) The weighted perfect domination problem. Inf Proc Lett 35:295–299

    Article  MathSciNet  Google Scholar 

  38. Yen CC (1992) Algorithmic aspects of perfect domination. PhD thesis, Department of Computer Science, National Tsing Hua University, Taiwam

  39. Yen CC, Lee RCT (1996) The weighted perfect domination problem and its variants. Discr Appl Math 66:147–160

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Information Engineering, National Cheng Kung University, No. 1, Ta-Hsueh Road, Tainan, 701, Taiwan

    Sun-Yuan Hsieh

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  1. Sun-Yuan Hsieh
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Correspondence to Sun-Yuan Hsieh.

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Hsieh, SY. An efficient parallel strategy for the perfect domination problem on distance-hereditary graphs. J Supercomput 39, 39–57 (2007). https://doi.org/10.1007/s11227-006-0003-6

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