Unique basis graphs

A set W ⊆ V (G) is called a resolving set, if for each two distinct vertices u, v ∈ V (G) there exists w ∈ W such that d(u, w) 6= 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.

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