On minimal doubly resolving sets in graphs

TL;DR

This paper investigates doubly resolving sets in graphs, providing bounds, exact values for important graphs, and characterizing graphs with maximum minimal doubly resolving set size.

Contribution

It introduces bounds on the size of doubly resolving sets, computes these sets for key graphs, and characterizes graphs with the largest minimal doubly resolving set.

Findings

Upper bound for 43D(G) based on order and diameter

Exact 43D(G) for specific important graphs

Characterization of graphs with 43D(G)=n-1

Abstract

Two vertices u,v of connected graph G are doubly resolved by x,y\in V(G)if d(v; x)-d(u; x)\neq d(v; y)-d(u; y): A set W of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of W. \psi(G) is the minimum cardinality of a doubly resolving set for the graph G. The aim of this paper is to investigate doubly resolving sets in graphs. An upper bound for \Psi(G) is obtained in terms of order and diameter of G. \psi (G) is computed for some important graphs and all graphs G of order n with the property \psi(G)=n-1 are characterized.