Closed 2-cell embeddings of graphs with no \(V_8\)-minors (Q5931420)

A closed 2-cell imbedding of a connected graph \(G\) in a surface (a closed 2-manifold) has every face bounded by a cycle of \(G\). The strong imbedding conjecture claims that every 2-connected graph \(G\) has a closed 2-cell imbedding in some surface. The authors show that the conjecture holds for 2-connected graphs \(G\) not containing \(V_8\) as a minor, where \(V_8\), the Möbius 4-ladder, can be described as the Cayley graph for \(Z_8\), generated by 1 and 4. That each such graph has a cycle double cover is then an immediate corollary. The proof of the theorem relies heavily on a classification of internally-4-connected graphs with no \(V_8\) minor, due to \textit{A. K. Kel'mans} [Sov. Math., Dokl. 33, 698-703 (1986; Zbl 0622.05043); translation from Dokl. Akad. Nauk SSSR 288, 531-535 (1986)] and, independently, to N. Robertson (preprint).