I am looking for a list of necessary and sufficient conditions for two loops on a (compact connected orientable) surface to be homotopic that could be made into a purely combinatorial definition of homotopy in embeddable (but not necessarily embedded) graphs (esp. triangulations of the surface). Or is there no chance for such a definition because homotopy is an essentially continuous concept, and graphs are discrete?
If two loops intersect transversally in exactly one point they are not homotopic to each other and not homotopic to a contractible loop.
If two loops - each on its own - disconnect the surface they are both contractible and thus homotopic to each other.
If two loops are shorter than the systole of the surface they are both contractible and thus homotopic to each other.
(The latter corresponds to a trivial sufficient condition for two cycles of a triangulation to be homotopic: if they are triangles.)