Intersection number and the stability of some inscribable graphs

TL;DR

This paper investigates the stability of inscribable graphs, which are graphs that can be realized as polyhedra inscribed in spheres, by combining Teichmüller theory and differential topology.

Contribution

It introduces a new approach to analyze the stability of inscribable graphs using Teichmüller theory and differential topology methods.

Findings

Identifies conditions under which inscribable graphs are stable.

Provides a framework for approximating inscribed polyhedra on convex surfaces close to spheres.

Advances understanding of inscribability and stability in polyhedral graph theory.

Abstract

A planar graph is inscribable if it is combinatorial equivalent to the skeleton of a polyhedra which is inscribed in a sphere. For an inscribable graph, in its combinatorial equivalent class, if we could always find polyhedra inscribed in any given convex surface which is sufficiently close to the sphere, then we call such an inscribable graph a stable one. By combining the Teichm\"{u}ller theory of packings with differential topology method, in this paper there is investigation on the stability of some inscribable graphs.