TITLE: Obstructions for embedding a graph on a surface of genus g
QUESTION [21 upvotes]: Kuratowski's theorem tells us the complete graph $K_5$ and the bipartite graph $K_{3,3}$ are the only obstructions to a graph being planar, ie embeddable in the plane with no edge-crossings.

Is the list of obstructions to being able to embed a graph with no edge-crossings on the surface of genus $g$ known to be finite for all $g$?

REPLY [5 votes]: For the projective plane, i.e. the nonorientable surface of genus 1, this is known. Look at "A Kuratowski Theorem for the Projective Plane" in the homepage of Dan Archdeacon here:
http://www.emba.uvm.edu/~archdeac/
This was his PhD thesis. In particular, he found that there are exactly 103 graphs such that if any graph $G$ contains one of these graphs as a subgraph, then $G$ cannot be not embedded into projective plane. You can refer to his original thesis for the list of the 103 graphs, or you can refer to the Appendix A of the book "Graphs on Surfaces" by Bojan Mohar and Carsten Thomassen.