Symbolic Constraints in Polyhedral Enclosure and Tetrahedral Decomposition in Genus-0 Polyhedra

TL;DR

This paper introduces a symbolic, coordinate-free framework for analyzing genus-zero polyhedra, enabling rapid feasibility checks of polyhedral configurations based solely on combinatorial data, reducing geometric validation efforts.

Contribution

The work provides exact combinatorial identities and bounds for genus-zero polyhedra, facilitating efficient pre-screening of feasible configurations without geometric embedding.

Findings

Exact incidence identities linking V, E, F to flatness parameter S

Parity-sensitive extremal behavior in E, F, S due to vertex-degree constraints

Linear ranges for tetrahedral decompositions with no interior vertices

Abstract

I present a coordinate-free, symbolic framework for determining whether a given set of polygonal faces can form a closed, genus-zero polyhedral surface and for predicting admissible internal tetrahedral decompositions consistent with incidence constraints. The method uses only discrete combinatorial variables, such as the number of tetrahedra T, internal gluing triangles Ni, and internal triangulation segments Si, and applies feasibility checks prior to any geometric embedding. For polyhedra in normal form, I record exact incidence identities linking V, E, and F to a flatness parameter S defined as the sum over faces of (degree minus three), and identify parity-sensitive extremal behavior in E, F, and S arising from minimal vertex-degree constraints. These external identities and parity-dependent bounds hold for genus-zero polyhedral graphs under standard simplicity and connectivity assumptions. For internal quantities, I prove exact relations Ni = 2T - V + 2 and T - Ni + Si = 1, and derive restricted linear ranges for tetrahedral decompositions in normal form with no interior vertices. Together, these results yield a symbolic workflow for rapid pre-screening of combinatorially impossible configurations, reducing reliance on costly geometric validation in computational geometry, graphics, and automated modeling.