Abstract
Let T = (V, E) be a tree. A core of T is a path P, for which the sum of the weighted distances from all vertices to this path is minimized. In this paper, we consider the semi-obnoxious case in which the vertices have positive or negative weights. We prove that, when the sum of the weights of vertices is negative, the core must be a single vertex and that, when the sum of the vertices’ weights is zero there exists a core that is a vertex. Morgan and Slater (J Algorithms 1:247–258, 1980) presented a linear time algorithm to find the core of a tree with only positive weights of vertices. We show that their algorithm also works for semi-obnoxious problems.
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Zaferanieh, M., Fathali, J. Finding a core of a tree with pos/neg weight. Math Meth Oper Res 76, 147–160 (2012). https://doi.org/10.1007/s00186-012-0394-5
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DOI: https://doi.org/10.1007/s00186-012-0394-5