Intersection number and the stability of some inscribable graphs

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.