Unique basis graphs

Abstract

A set W C V(G) is called a resolving set, if for each two distinct vertices u v € V(G) there exists w € W such that d(uw) d(v,w), where d(x,y) is the distance between the vertices x and y. A resolving set for G with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a unique basis graph. In this paper, we study some properties of unique basis graphs.