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

Abstract

A subset Δ of non-negative integers N 0 is called a numerical semigroup if it is a submonoid of N 0 and has a finite complement in N 0 . A graph G Δ is called a Δ ( α , β ) -graph if there exists a numerical semigroup Δ with multiplicity α and embedding dimension β such that V ( G Δ ) = { v i : i ∈ N 0 ∖ Δ } and E ( G Δ ) = { v i v j ⇔ 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.