Classification of Planar Graphs Associated to the Ideal of the Numerical Semigroup

TL;DR

This paper characterizes when the graphs associated with irreducible ideals of numerical semigroups are planar, providing a complete classification up to isomorphism and identifying non-planar cases.

Contribution

It offers a complete characterization of the planarity of graphs derived from irreducible ideals in numerical semigroups, a novel classification in this area.

Findings

Identifies conditions for planarity of $G_I(\Lambda)$

Classifies all such graphs up to isomorphism

Determines which irreducible ideals lead to non-planar graphs

Abstract

Let $\Lambda$ be a numerical semigroup and $I\subset \Lambda$ be an ideal of $\Lambda$. The graph $G_I(\Lambda)$ assigned to an ideal $I$ of $\Lambda$ is a graph with elements of $(\Lambda \setminus I)^*$ as vertices and any two vertices $x,y$ are adjacent if and only if $x+y \in I$. In this paper we give a complete characterization (up to isomorphism ) of the graph $G_I(\Lambda)$ to be planar, where $I$ is an irreducible ideal of $\Lambda$. This will finally characterize non planar graphs $G_I(\Lambda)$ corresponding to irreducible ideal $I$.