an efficient algorithm for finding the semi-obnoxious $(k,l)$-core of a tree

in this paper we study finding the $(k,l)$-core problem on a tree which the vertices have positive or negative weights. let $t=(v,e)$ be a tree. the $(k,l)$-core of $t$ is a subtree with at most $k$ leaves and with a diameter of at most $l$ which the sum of the weighted distances from all vertices to this subtree is minimized. we show that, when the sum of the weights of vertices is negative, the $(k,l)$-core must be a single vertex. then we propose an algorithm with time complexity of $o(n^2log n)$ for finding the $(k,l)$-core of a tree with pos/neg weight, which is in fact a modification of the one proposed by becker et al. [networks 40 (2002) 208].