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

Abstract Let T=(V,E) be a tree graph with non-negative costs defined on the vertices. A vertex τ is called a separating vertex for u and v if the distances of τ to u and v are not equal. A set of vertices L ⊆ V is a feasible solution for the non-landmarks model (NL), if for every pair of distinct vertices, u,v ∈ V\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.