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

Abstract

A subset Δ of non-negative integers [Formula: see text] is called a numerical semigroup if it is a submonoid of [Formula: see text] and has a finite complement in [Formula: see text]. A graph [Formula: see text] is called a [Formula: see text]-graph if there exists a numerical semigroup Δ with multiplicity α and embedding dimension β such that [Formula: see text] and [Formula: see text]. In this article, we compute the [Formula: see text]-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 [Formula: see text]-graphs respectively, whose metric dimension is 2.

Keywords

• Numerical semigroup
• metric dimension
• Genus
• Frobenius number
• 05C25
• 16U60