On smallest $3$-polytopes of given graph radius

TL;DR

This paper investigates the smallest 3-polytopes with a given graph radius, confirming under certain conditions that the 2(r-1)-gonal prism is unique in this regard.

Contribution

It provides a partial positive answer to the classical question about the uniqueness of minimal 3-polytopes with specified graph radius under additional assumptions.

Findings

The 2(r-1)-gonal prism is the unique smallest 3-polytope of radius r under certain conditions.

The paper establishes conditions where this uniqueness holds.

It advances understanding of the structure of 3-polytopes with given graph radius.

Abstract

The $3$-polytopes are planar, $3$-connected graphs. A classical question is, for $r\geq 3$, is the $2(r-1)$-gonal prism $K_2\times C_{2(r-1)}$ the unique $3$-polytope of graph radius $r$ and smallest size? Under some extra assumptions, we answer this question in the positive.