The weighted 2-metric dimension of trees in the non-landmarks model

Let T=(V,E) be a tree graph with non-negative weights defined on the vertices. A vertex z is called a separating vertex for u and v if the distances of z to u and v are not equal. A set of vertices L\subseteq V is a feasible solution for the non-landmarks model (NL), if for every pair of distinct vertices, u,v \in V\setminus L, there are at least two vertices of L separating them. Such a feasible solution is called a "landmark set". We analyze the structure of landmark sets for trees and design a linear time algorithm for finding a minimum cost landmark set for a given tree graph.