On unicyclic graphs of metric dimension 2 with vertices of degree 4

Abstract

We show that if \(G\) is a unicyclic graph with metric dimension \(2\) and \(\{a,b\}\) is a metric basis of \(G\) then the degree of any vertex \(v\) of \(G\)  is at most \(4\) and degrees of both \(a\) and \(b\) are at most \(2\). The constructions of unispider and semiunispider graphs  and their knittings are introduced. Using these constructions all  unicyclic graphs of metric dimension \(2\) with vertices of degree  \(4\) are characterized.