Commun. Korean Math. Soc. 2018; 33(2): 631-638
Online first article January 8, 2018 Printed April 30, 2018
https://doi.org/10.4134/CKMS.c170174
Copyright © The Korean Mathematical Society.
Xue-Gang Chen, Moo Young Sohn
North China Electric Power University, Changwon National University
Chellai et al.~\cite{1} 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 ~\cite{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.
Keywords: $[1, 2]$-domination number, tree, polynomial algorithm
MSC numbers: 05C69, 05C85
2018; 33(1): 361-369
2012; 27(4): 665-668
2010; 25(4): 655-669
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