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

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 $(\Lambda \setminus I{)}^{\ast }$ 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 $I$.