Consecutive non-orientable embedding obstructions

Definitions and mathematical statement

For an integer $g\ge1$, let $N_g$ be the connected closed nonorientable surface that is the connected sum of $g$ projective planes. A finite graph embeds in $N_g$ if its vertices can be placed at distinct points and its edges drawn as arcs meeting only at common endpoints. Embeddings are not required to be cellular.

A minor of a finite simple graph is obtained by deleting vertices or edges and contracting edges, subsequently discarding loops and merging parallel edges. A proper minor is one not isomorphic to the original graph. Call $G$ a minor-minimal obstruction for $N_g$ when $G$ does not embed in $N_g$ but every proper minor of $G$ does.

The question asks whether there exist a finite graph $G$ and an integer $g\ge1$ such that $G$ is a minor-minimal obstruction for both $N_g$ and $N_{g+1}$. Equivalently, one requires \[ G\not\hookrightarrow N_{g+1},\qquad H\hookrightarrow N_g\quad\text{for every proper minor }H\text{ of }G. \] Embeddability in $N_g$ implies embeddability in $N_{g+1}$, so these conditions give both obstruction properties.

This consecutive-surface form also captures the wording asking for any two distinct nonorientable surfaces. If a graph is an obstruction for $N_i$ and $N_j$ with $i<j$, monotonicity makes it an obstruction for each intervening surface, and in particular for $N_i,N_{i+1}$. Thus no particular genus is fixed in advance, and the existence quantifier over $g$ is essential.

Short English statement

Can a single finite graph be a forbidden minor for two consecutive nonorientable surfaces?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3327 (OPG-172), OpenGarden. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Consecutive non-orientable embedding obstructions. Original problem and discussion. http://www.openproblemgarden.org/op/consecutive_non_orientable_embedding_obstructions.