Metric Dimensions of Bicyclic Graphs

The distance d ( v a , v b ) between two vertices of a simple connected graph G is the length of the shortest path between v a and v b . Vertices v a , v b of G are considered to be resolved by a vertex v if d ( v a , v ) ≠ d ( v b , v ) . An ordered set W = { v 1 , v 2 , v 3 , … , v s } ⊆ V ( G ) is said to be a resolving set for G , if for any v a , v b ∈ V ( G ) , ∃ v i ∈ W ∋ d ( v a , v i ) ≠ d ( v b , v i ) . The representation of vertex v with respect to W is denoted by r ( v | W ) and is an s -vector( s -tuple) ( d ( v , v 1 ) , d ( v , v 2 ) , d ( v , v 3 ) , … , d ( v , v s ) ) . Using representation r ( v | W ) , we can say that W is a resolving set if, for any two vertices v a , v b ∈ V ( G ) , we have r ( v a | W ) ≠ r ( v b | W ) . A minimal resolving set is termed a metric basis for G . The cardinality of the metric basis set is called the metric dimension of G , represented by d i m ( G ) . In this article, we study the metric dimension of two types of bicyclic graphs. The obtained results prove that they have constant metric dimension.