Strong metric dimension of rooted product graphs

Abstract

Let G be a connected graph. A vertex w strongly resolves two different vertices of G if there exists a shortest path, which contains the vertex v or a shortest path, which contains the vertex u. A set W of vertices is a strong metric generator for G if every pair of different vertices of G is strongly resolved by some vertex of W. The smallest cardinality of a strong metric generator for G is called the strong metric dimension of G. It is known that the problem of computing this invariant is NP-hard. According to that fact, in this paper we study the problem of computing exact values or sharp bounds for the strong metric dimension of the rooted product of graphs and express these in terms of invariants of the factor graphs.