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Tractable connected domination for restricted bipartite graphs

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Abstract

Connected domination, i.e. the problem of finding a minimum connected dominating set in a graph, was known to be \(\mathcal {NP}\)-complete for chordal bipartite graphs, but to be tractable for convex bipartite graphs. In this paper, connected domination is shown to be tractable for circular- and triad-convex bipartite graphs respectively, by efficient reductions from these graphs to convex bipartite graphs.

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References

  • Akhbari MH, Hasni R, Favaron O, Karami H, Sheikholeslami SM (2013) On the outer-connected domination in graphs. J Comb Optim 26:10–18

    Article  MATH  MathSciNet  Google Scholar 

  • Damaschke P, Müller H, Kratsch D (1990) Domination in convex and chordal bipartite graphs. Inf Process Lett 36(5):231–236

    Article  MATH  Google Scholar 

  • Du H, Wu W, Shan S, Kim D, Lee W (2012) Constructing weakly connected dominating set for secure clustering in distributed sensor network. J Comb Optim 23:301–307

    Article  MATH  MathSciNet  Google Scholar 

  • Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W.H.Freeman and Company, New York

    MATH  Google Scholar 

  • Grover F (1967) Maximum matching in a convex bipartite graph. Nav Res Logist Q 14:313–316

    Article  Google Scholar 

  • Jiang W, Liu T, Ren TN, Xu K (2011a) Two hardness results on feedback vertex sets. In: Proceedings of of FAW-AAIM, pp 233–243

  • Jiang W, Liu T, Wang C, Xu K (2013) Feedback vertex sets on restricted bipartite graphs. Theor Comput Sci 507(7):41–51

    Article  MATH  MathSciNet  Google Scholar 

  • Jiang W, Liu T, Xu K (2011b) Tractable feedback vertex sets in restricted bipartite graphs. In: Proceedings of COCOA, pp 424–434

  • Li Y, Wu Y, Ai C, Beyah R (2012) On the construction of \(k\)-connected \(m\)-dominating sets in wireless networks. J Comb Optim 23:118–139

    Article  MATH  MathSciNet  Google Scholar 

  • Liang YD, Blum N (1995) Circular convex bipartite graphs: maximum matching and Hamiltonian circuits. Inf Process Lett 56:215–219

    Article  MATH  MathSciNet  Google Scholar 

  • Lu M, Liu T, Xu K (2013a) Independent domination: reductions from circular- and triad-convex bipartite graphs to convex bipartite graphs. In: Proceedings of FAW-AAIM, pp 142–152

  • Lu Z, Liu T, Xu K (2013b) Tractable connected domination for restricted bipartite graphs (Extended Abstract). In: Proceedings of COCOON, pp 721–728

  • Lu Z, Lu M, Liu T, Xu K (2013c) Circular convex bipartite graphs: feedback vertex set. In: Proceedings of COCOA, pp 272–283

  • Müller H, Brandstät A (1987) The NP-completeness of Steiner tree and dominating set for chordal bipartite graphs. Theor Comput Sci 53(2–3):257–265

    Article  MATH  Google Scholar 

  • Pfaff J, Laskar R, Hedetniemi ST (1983) NP-completeness of total and connected domination, and irredundance for bipartite graphs. Technical Report 428, Department of Mathematical Sciences, Clemenson University

  • Sampathkumar E, Walikar HB (1979) The connected domination number of a graph. Math Phys Sci 13(6):607–613

    MATH  MathSciNet  Google Scholar 

  • Shang W, Yao F, Wan P, Hu X (2008) On minimum \(m\)-connected \(k\)-dominating set problem in unit disc graphs. J Comb Optim 16:99–106

    Article  MATH  MathSciNet  Google Scholar 

  • Song Y, Liu T, Xu K (2012) Independent domination on tree convex bipartite graphs. In: Proceedings of FAW-AAIM, pp 129–138

  • Wang C, Liu T, Jiang W, Xu K (2012) Feedback vertex sets on tree convex bipartite graphs. In: Proceedings of COCOA, pp 95–102

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Acknowledgments

Thanks are due to Professor Kaile Su for encouragements and supports, to Professor Francis Y. L. Chin for bringing our attention to the notion of circular convex bipartite graphs during FAW-AAIM 2011, and to anonymous referees whose comments are very helpful to improve our presentations. Partially supported by National 973 Program of China (Grant No. 2010CB328103), Natural Science Foundation of China (Grant Nos. 61370052 and 61370156), and State Key Laboratory of Software Development Environment Open Fund (Grant No. SKLSDE-2012KF-06).

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Authors and Affiliations

  1. Key Laboratory of High Confidence Software Technologies, Ministry of Education, Institute of Software, School of Electronic Engineering and Computer Science, Peking University, Beijing , 100871, China

    Tian Liu & Zhao Lu

  2. National Lab of Software Development Environment, Beihang University, Beijing , 100191, China

    Ke Xu

Authors
  1. Tian Liu
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  2. Zhao Lu
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  3. Ke Xu
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Correspondence to Tian Liu or Ke Xu.

Additional information

A preliminary version of this paper appeared as an extended abstract in Proc. of COCOON 2013.

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Liu, T., Lu, Z. & Xu, K. Tractable connected domination for restricted bipartite graphs. J Comb Optim 29, 247–256 (2015). https://doi.org/10.1007/s10878-014-9729-x

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  • DOI: https://doi.org/10.1007/s10878-014-9729-x

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