Obstructions for two-vertex alternating embeddings of graphs in surfaces

A class of graphs that lies strictly between the classes of graphs of genus (at most) k−1k−1 and kk is studied. For a fixed orientable surface SkSk of genus kk, let Axyk be the minor-closed class of graphs with terminals xx and yy that either embed into Sk−1Sk−1 or admit an embedding ΠΠ into SkSk such that there is a ΠΠ-face where xx and yy appear twice in the alternating order. In this paper, the obstructions for the classes Axyk are studied. In particular, the complete list of obstructions for Axy1 is presented.