Unique Basis Graphs

Abstract

A set \(W \subseteq V(G)\) is called a resolving set, if for each two distinct vertices \(u, v \in V(G)\) there exists \(w \in W\) such that \(d(u, w) \neq d(v, w)\), 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.