Further new results on strong resolving partitions for graphs

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.