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

Let T=(V,E)T=(V,E) be a tree graph with non-negative costs defined on the vertices. A vertex ττ is called a separating vertex for uu and vv if the distances of ττ to uu and vv are not equal. A set of vertices L⊆VL⊆V is a feasible solution for the non-landmarks model (NL), if for every pair of distinct vertices, u,v∈V∖Lu,v∈V∖L, there are at least two vertices of LL 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.