A classification of polyhedral graph by combinatorially rigid vertices

TL;DR

This paper introduces a classification scheme for polyhedral graphs based on the concept of combinatorially rigid vertices, focusing on vertices with limited non-triangular faces, and explores their properties and implications.

Contribution

It defines a new classification criterion for polyhedral graphs using combinatorial rigidity and proposes a conjecture related to the distribution of rigid vertices.

Findings

Defined combinatorial rigidity for vertices with up to three non-triangular faces.

Studied the distribution and properties of rigid vertices in polyhedra.

Proposed a conjecture on the classification of polyhedra based on rigid vertices.

Abstract

When the number of non-triangular faces adjacent to a vertex $v$ is less than or equal to three, the vertex $v$ will be called (\emph{combinatorially}) \emph{rigid}. We study the number of rigid vertices and suggest a conjecture on a classification of polyhedra.