It is easy to see that every planar graph is a minor of another planar graph of maximum degree 3. Georgakopoulos proved that every finite K5-minor free graph is a minor of another K5-minor-free graph of maximum degree 22, and inquired if this is smallest possible. This motivates the following generalization: Let 𝐶 be a minor-closed class. What is the minimum 𝑘 such that any graph in 𝐶 is a minor of a graph in 𝐶 of maximum degree 𝑘? Denote the minimum by Δ(𝐶) and set it to be ∞ if no such 𝑘 exists.
We explore the value of Δ(𝐶) for various minor closed classes, and eventually prove that a minor-closed class 𝐶 excludes an apex graph if and only if there exists a proper minor-closed superclass 𝐶′ of 𝐶 with Δ(𝐶′) = 3 if and only if there exists a proper minor-closed superclass 𝐶′ of 𝐶 with finite Δ(𝐶′). This complements a list of 5 other characterizations of the minor-closed classes excluding an apex graph by Dujmovic, Morin and Wood.
Furthermore, we extend and simplify Markov and Shi’s result that not every graph of treewidth ≤ k has a degree 3 expansion of treewidth ≤ 𝑘. Finally, we simplify Georgakopoulos’ proof on the existence of a countable universal graph of 𝐹 𝑜𝑟𝑏(𝐾5).