Scaffold for the polyhedral embedding of cubic graphs

TL;DR

This paper establishes a one-to-one correspondence between extended graphs and polyhedral embeddings of cubic graphs, providing a new framework for understanding their embedding structures.

Contribution

It introduces the concept of extended graphs for cubic graphs and proves their correspondence with polyhedral embeddings, advancing the theoretical understanding of graph embeddings.

Findings

Extended graphs are constructed by adding edges corresponding to facial subwalks.

There is a one-to-one correspondence between extended graphs and polyhedral embeddings.

The framework aids in classifying and analyzing embeddings of cubic graphs.

Abstract

Let $G$ be a cubic graph and $\Pi$ be a polyhedral embedding of this graph. The extended graph, $G^{e},$ of $\Pi$ is the graph whose set of vertices is $V(G^{e})=V(G)$ and whose set of edges $E(G^{e})$ is equal to $E(G) \cup \mathcal{S}$, where $\mathcal{S}$ is constructed as follows: given two vertices $t_0$ and $t_3$ in $V(G^{e})$ we say $[t_0 t_3] \in \mathcal{S},$ if there is a $3$--path, $(t_0 t_1 t_2 t_3) \in G$ that is a $\Pi$-- facial subwalk of the embedding. We prove that there is a one to one correspondence between the set of possible extended graphs of $G$ and polyhedral embeddings of $G$.