Minor of graph:

deletion of edges and/or vertices; and contraction of edges to vertices

deletion of edges and/or vertices; and contraction of edges to vertices

Looking at families of graphs that are closed under taking minors [i.e. coarse-graining]

E.g. planarity

Examples:

Apex graph

Toroidal graph

Graphs that can be embedded on any fixed 2D manifold

Graphs of any fixed genus

For any family closed under minor taking, there is a finite set of forbidden minors

E.g. planarity

Examples:

Apex graph

Toroidal graph

Graphs that can be embedded on any fixed 2D manifold

Graphs of any fixed genus

For any family closed under minor taking, there is a finite set of forbidden minors

Conservation of genus under graph minoring....

If you can’t generate a forbidden minor, you stay e.g. in the genus

[[ In a first approximation, every forbidden minor corresponds to a particle ]]

If you can’t generate a forbidden minor, you stay e.g. in the genus

[[ In a first approximation, every forbidden minor corresponds to a particle ]]

https://en.wikipedia.org/wiki/Forbidden_graph_characterization

https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theorem

“Finite obstruction set for each genus”

https://mathoverflow.net/questions/46655/obstructions-for-embedding-a-graph-on-a-surface-of-genus-g

https://arxiv.org/pdf/1608.04066.pdf