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

TL;DR

This paper studies the minimum weighted landmark set problem in trees within the non-landmarks model, providing structural insights and a linear time algorithm for optimal solutions.

Contribution

It introduces a detailed structural analysis of landmark sets in trees and develops a linear time algorithm to find minimum cost landmark sets.

Findings

Structural characterization of landmark sets in trees

Linear time algorithm for minimum weighted landmark set

Efficient solution for non-landmarks model in trees

Abstract

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.