대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제33권2호
  • /
  • Pages.631-638
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  • 2018
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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ON [1, 2]-DOMINATION IN TREES

  • Chen, Xue-Gang (Department of Mathematics North China Electric Power University) ;
  • Sohn, Moo Young (Department of Mathematics Changwon National University)
  • 투고 : 2017.04.26
  • 심사 : 2017.11.07
  • 발행 : 2018.04.30
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초록

Chellai et al. [3] gave an upper bound on the [1, 2]-domination number of tree and posed an open question "how to classify trees satisfying the sharp bound?". Yang and Wu [5] gave a partial solution for tree of order n with ${\ell}$-leaves such that every non-leaf vertex has degree at least 4. In this paper, we give a new upper bound on the [1, 2]-domination number of tree which extends the result of Yang and Wu. In addition, we design a polynomial time algorithm for solving the open question. By using this algorithm, we give a characterization on the [1, 2]-domination number for trees of order n with ${\ell}$ leaves satisfying $n-{\ell}$. Thereby, the open question posed by Chellai et al. is solved.

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과제정보

연구 과제 주관 기관 : National Research Foundation of Korea(NRF)

참고문헌

  1. A. T. Amin and P. J. Slater, Neighborhood domination with parity restrictions in graphs, Congr. Numer. 91 (1992), 19-30.
  2. A. T. Amin, All parity realizable trees, J. Combin. Math. Combin. Comput. 20 (1996), 53-63.
  3. M. Chellali, T. W. Haynes, and S. T. Hedetniemi, [1, 2]-sets in graphs, Discrete Appl. Math. 161 (2013), no. 18, 2885-2893. https://doi.org/10.1016/j.dam.2013.06.012
  4. I. J. Dejter, Quasiperfect domination in triangular lattices, Discuss. Math. Graph Theory 29 (2009), no. 1, 179-198. https://doi.org/10.7151/dmgt.1439
  5. X. Yang and B. Wu, [1, 2]-domination in graphs, Discrete Appl. Math. 175 (2014), 79-86. https://doi.org/10.1016/j.dam.2014.05.035