On the Problem of Relative Components of Minimal Graphs

1935, D.König asked the question on page 199 in his famous book with the title “Theorie der endlichen und unendlichen Graphen” whether there exists a Kuratowski-type theorem for every closed orientable surface F_p, p ∈ N₀. The modern version of Kuratowski’s theorem is:The minimalbasis M₁(F₀) consists of exactly two graphs, the so-called Kuratowski graphs K_3,3 and K₅ with M₁(F₀) = {K_3,3,K₅}, where the minimalbasis M₁(F₀) is defined as the set of all >₁-minimal graphs of the set Γ(F₀) ⊂ Γ of all nonplanar graphs. Γ stands for the set of all finite undirected graphs without loops and multiple edges and >₁ is the well-known subdivision relation on Γ or Γ(F₀), respectively.