The Graph Minor Theorem of Robertson and Seymour asserts that any minor-closed graph property is determined by a finite set of forbidden graph minors. It is a broad generalization e.g. of the Kuratowski-Wagner theorem, which characterizes planarity in terms of two forbidden minors: the complete graph $K_5$ and the complete bipartite graph $K_{3,3}$.

Embeddability of a graph in the **projective plane** is such a minor-closed property as well, and it is known that there are 35 forbidden minors that characterize projective planarity. All 35 minors are known, a recent reference from 2012 is, for example, https://smartech.gatech.edu/bitstream/handle/1853/45914/Asadi-Shahmirzadi_Arash_201212_PhD.pdf.

A classical reference is *Graphs on Surfaces* from Mohar and Thomassen,
Johns Hopkins University Press 2001.

I am interested in the Colin de Verdière numbers for these 35 forbidden minors and have searched for them for a while now, but could not find anything.

Question:So I wondered whether the Colin de Verdière graph invariants for the whole set of these 35 forbidden minors are actually known? I would be grateful for any reference.

UPDATE:

Updating this question based on a great comment from Martin Winter. As he points out, the Colin de Verdière number $\mu$ is known and $\mu=4$ for a handful of these 35 forbidden minors, e.g. the disjoint unions of $K_5$ and $K_{3,3}$.

Interestingly, as outlined in his answer to a related question (Algebraic graph invariant $\mu(G)$ which links Four-Color-Theorem with Schrödinger operators: further topological characterizations of graphs?), it follows that the Colin de Verdière invariant **cannot** provide a full characterization of graph embeddings e.g. in the projective plane.

notfor all forbidden minors, then we can't have "projective planarity $\Leftrightarrow$ $\mu \le 5$". So the hope is thatallof them have $\mu$ larger five, or am I missing something? $\endgroup$