Obstructions for embedding graphs into surfaces

Only for two surfaces, the 2-sphere and the projective plane, the complete list of obstructions is known. We aim to expand our understanding of obstructions for higher-genus surfaces by studying obstructions of low connectivity. Classes of graphs are described such that each obstruction of connectivity 2 is obtained as a 2-sum of graphs from those classes. In particular, this structure allows us to determine the complete lists of obstructions of connectivity 2 for the torus and the Klein bottle.