On Doubly Resolving Sets in Graphs

Abstract

Two vertices u, v in a connected graph G are doubly resolved by vertices x, y of G if d(v,x)-d(u,x)≠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. Doubly resolving number of a graph G, denoted by ψ(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 ψ(G) is obtained in terms of order and diameter of G. ψ(G) is computed for some graphs, and all graphs G of order n with the property ψ(G) = n- 1 are determined. Also, doubly resolving sets for unicyclic graphs are studied and it is proved that the difference between the number of leaves and doubly resolving number of a unicyclic graph is at most 2. © 2022, This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply.