On the metric dimension of graphs associated with irreducible and Arf numerical semigroups

A subset Δ of non-negative integers N0 is called a numerical semigroup if it is a submonoid of N0 and has a finite complement in N0. A graph GΔ is called a Δ(α,β)-graph if there exists a numerical semigroup Δ with multiplicity α and embedding dimension β such that V(GΔ)={vi:i∈N0∖Δ} and E(GΔ)={vivj⇔i+j∈Δ}. In this article, we compute the Δ(α,β)-graphs for irreducible and Arf numerical semigroups having a metric dimension of 2. It is proved that if Δ be an irreducible and arf numerical semigroup then there are exactly 2 and 8 non-isomorphic Δ(α,β)-graphs respectively, whose metric dimension is 2.