Realizability of singular levels of Morse functions by unions of geodesics

TL;DR

This paper investigates which simple graphs of degree 4 with up to three vertices can be realized as unions of closed geodesics on surfaces of constant curvature, linking graph theory with geometric realizability.

Contribution

It identifies specific graphs that can be represented by unions of geodesics on certain surfaces, bridging combinatorial graph properties with geometric realizability.

Findings

Identified special degree-4 graphs with up to 3 vertices

Determined which graphs can be represented by geodesic unions

Connected graph realizability to surface geometry

Abstract

We list special graphs of degree 4 with at most 3 vertices (atoms from the theory of integrable hamiltonian systems) which could be represented by a union of closed geodesics on the one of the following surfaces with metric of constant curvature: sphere, projective plane, torus, Klein bottle.