Scaffold for the polyhedral embedding of cubic graphs

Abstract

Let G be a cubic graph and Π be a polyhedral embedding of this graph. The extended graph, G e , of Π 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) ∪ S, where S is constructed as follows: given two vertices t 0 and t 3 in V (G e ) we say [t 0 t 3 ] ∈ S, if there is a 3-path, (t 0 t 1 t 2 t 3 ) ∈ G that is a Π-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.