Abstract
Abstract A set W of vertices of a connected graph G strongly resolves two different vertices x, y ∉ W if either d G (x, W) = d G (x, y) + d G (y, W) or d G (y, W) = d G (y, x) + d G (x, W), where d G (x, W) = min{d(x,w): w ∈ W} and d(x,w) represents the length of a shortest x − w path. An ordered vertex partition Π = {U 1, U 2,…,U k } of a graph G is a strong resolving partition for G, if every two different vertices of G belonging to the same set of the partition are strongly resolved by some other set of Π. The minimum cardinality of any strong resolving partition for G is the strong partition dimension of G. In this article, we obtain several bounds and closed formulae for the strong partition dimension of some families of graphs and give some realization results relating the strong partition dimension, the strong metric dimension and the order of graphs.
Highlights
Given a connected graph G, a vertex v ∈ V(G) distinguishes two distinct vertices x, y ∈ V(G), if dG(v, x) ≠ dG (v, y), where dG(v, y) represents the length of a shortest x − y path
An equivalent terminology was introduced by Harary and Melter in [2], where metric generators were called resolving sets
The terminology of metric generators was first presented in [3], as a more natural way of understanding such structure. This latter terminology is arising from the theory of metric spaces
Summary
Given a connected graph G, a vertex v ∈ V(G) distinguishes two distinct vertices x, y ∈ V(G), if dG(v, x) ≠ dG (v, y), where dG(v, y) represents the length of a shortest x − y path. Metric generators were studied in [21], where the authors found an interesting connection between the strong metric bases of a graph and the vertex cover number of a related graph which they called “strong resolving graph”. The study of the strong metric dimension of a graph is closely related to the study of the vertex cover number of a related graph, known as strong resolving graph [21] To introduce such a graph, we need the following terminology. The strong resolving graph of a graph G is a graph GSR with vertex set V(GSR) = ∂(G), where two vertices u,v are adjacent in GSR if and only if u and v are mutually maximally distant in G. For more information on the strong resolving graph of a graph (as a proper graph operation), we suggest a previous study [23], where several structural properties of such graphs were studied, and the recent work [24] which gives some new results concerning strong resolving graphs
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