On unicyclic graphs of metric dimension \(2\)

Abstract

A metric basis \(S\) of a graph \(G\) is the subset  of vertices of minimum cardinality  such that all other vertices are uniquely determined by their distances to the vertices in \(S\). The metric dimension of a graph \(G\) is the  cardinality of the subset \(S\).  A  unicyclic graph is a graph containing exactly one cycle. The construction of a  knitting unicyclic graph is introduced. Using this construction all unicyclic graphs with two main vertices  and  metric dimensions  \(2\) are   characterized.