Can anyone help me with this one?
Let there be a torus with a graph embedding (I hope I'm using the right term). By embedding I mean that the surface of the torus is divided in multiple triangular faces whose nodes and sides form the graph.
The torus has a tree-cotree decomposition. The cotree is the sub-graph that contains all the nodes of the initial graph and all the edges that are not in the tree.
Each edge of the cotree forms a unique loop with the branches of the tree. The loops are of three kinds:
- $\alpha$-loops are the simplest ones: they do not surround the hole or the torus itself
- $\beta$-loops surround the torus once (maybe more than once?)
- $\gamma$-loops surround the torus hole once (maybe more than once?)
Given only the tree-cotree decomposition is there a an algorithm which can tell the type of loop each cotree edge generates?
Can this method be extended for other multiple connected domains (like a double torus)?
If not, which types of mathematics can help me better understand (and possibly solve) this problem? Of course graph theory is essential, but I suspect topology and abstract algebra are also useful. However these are huge part of mathematics and I really have no clue where to start.