Abstract
Let T be a weighted tree with a positive number w(v) associated with each vertex v. A subtree S is a w-central subtree of the weighted tree T if it has the minimum eccentricity \(e_L(S)\) in median graph \(G_{LW}\). A w-central subtree with the minimum vertex weight is called a least w-central subtree of the weighted tree T. In this paper we show that each least w-central subtree of a weighted tree either contains a vertex of the w-centroid or is adjacent to a vertex of the w-centroid. Also, we show that any two least w-central subtrees of a weighted tree either have a nonempty intersection or are adjacent.
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References
Bielak H, Pańczyk M (2012) A self-stabilizing algorithm for finding weighted centroid in trees. Ann UMCS Inform AI XII 2:27–37
Brandeau ML, Chiu SS (1989) An overview of representative problem in location research. Manage Sci 35:645–674
Hakimi SL (1964) Optimal locations of switching centers and the absolute centers and medians of a graph. Oper Res 12:450–459
Hamina M, Peltola M (2011) Least central subtrees, center, and centroid of a tree. Networks 57:328–332
Kariv O, Hakimi SL (1979) An algorithm approach to network location problems. II: the \(p\)-medians. SIAM J Appl Math 37:539–560
Nieminen J, Peltola M (1999) The subtree center of a tree. Networks 34:272–278
Ried KB (1991) Centroids to center in trees. Networks 21:11–17
Smart C, Slater PJ (1999) Center, median, and centroid subgraphs. Networks 34:303–311
Spoerhase J, Wirth H-C (2009) \((r, p)\)-centroid problems on paths and trees. Theoret Comput Sci 410:5128–5137
Tamir A (1988) Improved complexity bounds for center location problems on networks by using dynamic data structures. SIAM J Discrete Math 1:377–396
Tansel BC, Francis RL, Lowe TJ (1983) Location on networks: a survey-Part I: the \(p\)-center and \(p\)-median problems. Manage Sci 29:482–497
Acknowledgements
This work was partially supported by the National Nature Science Foundation of China (Grant Numbers: 11471210, 11571222). We are very thankful to the referees for their careful reading of this paper and all helpful comments.
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Shan, E., Kang, L. The w-centroids and least w-central subtrees in weighted trees. J Comb Optim 36, 1118–1127 (2018). https://doi.org/10.1007/s10878-017-0174-5
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DOI: https://doi.org/10.1007/s10878-017-0174-5