Graph-truncations of $3$-polytopes

TL;DR

This paper investigates how truncating edges along a graph affects the simplicity and flag properties of 3-polytopes, providing criteria and extending realization theorems to flag polytopes.

Contribution

It introduces criteria for when edge truncations preserve simplicity and flag properties, and extends Eberhard's theorem to flag 3-polytopes.

Findings

Criteria for simple and flag polytopes after truncation

Extension of Eberhard's theorem to flag polytopes

Conditions for realizing polygon vectors in flag 3-polytopes

Abstract

In this paper we study the operation of cutting off edges of a simple $3$-polytope $P$ along the graph $\Gamma$. We give the criterion when the resulting polytope is simple and when it is flag. As a corollary we prove the analog of Eberhard's theorem about the realization of polygon vectors of simple $3$-polytopes for flag polytopes.