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On unigraphic $3$-polytopes of radius one
Combinatorics · Submitted 5 Jul 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We give an exhaustive list of these sequences for the case where one degree equals $p-1$ and exactly two or three of them equal $3$.
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The abstract, sentence by sentence
We ask which degree sequences admit a unique realisation as a $3$-polytopal graph (polyhedron) on $p$ vertices. We give an exhaustive list of these sequences for the case where one degree equals $p-1$ and exactly two or three of them equal $3$. We also find all $3$-polytopes of radius one with $p\leq 17$, and those with $q\leq 41$ edges, by developing a fast algorithm and making use of High Performance Computing.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We ask which degree sequences admit a unique realisation as a $3$-polytopal graph (polyhedron) on $p$ vertices. | background 0.87 | 0.30 | 0.05 | unclear 0.87 |
| s1 | We give an exhaustive list of these sequences for the case where one degree equals $p-1$ and exactly two or three of them equal $3$. | result 0.71 | 0.96 | 0.46 | asserted 1.00 |
| s2 | We also find all $3$-polytopes of radius one with $p\leq 17$, and those with $q\leq 41$ edges, by developing a fast algorithm and making use of High Performance Computing. | contribution 0.62 | 0.97 | 0.16 | asserted 1.00 |
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The paper
Original abstract
We ask which degree sequences admit a unique realisation as a $3$-polytopal graph (polyhedron) on $p$ vertices. We give an exhaustive list of these sequences for the case where one degree equals $p-1$ and exactly two or three of them equal $3$. We also find all $3$-polytopes of radius one with $p\leq 17$, and those with $q\leq 41$ edges, by developing a fast algorithm and making use of High Performance Computing.
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How the literature uses it
2 arXiv papers cite this in 20 sentences
Papers citing it · 16 sentences without a confident label
Builds on it 1 paper
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Firtly, it is straightforward to adapt the proof of [ref] to see that, when $a≥3$, $G$ has at least one cycle, with the only exception of $s: 5,4^2,3^3$ (where $p=6$).
Characterising $3$-polytopes of radius one with unique realisation · First part of the proof · 0.80
Compares with it 1 paper
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The sequence type C1 from [ref] is just the special case $a=3$ of type C from Theorem [ref], and the exceptional $5,4^2,3^3$ and $6,5^3,3^3$ from [ref] both satisfy $p≤3a-1$.
Characterising $3$-polytopes of radius one with unique realisation · Main Theorem · 0.87
What it cites
Cites as background 2 papers
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For recent results on degree sequences of $3$-polytopes, see e.g. [ref] .
Constructing certain families of $\mathbf{3}$-polytopal graphs · Results · 0.97
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For recent results on degree sequences of $3$-polytopes, see e.g. [ref] .
Self-dual polyhedra of given degree sequence · Results · 0.97
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