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) $k-1$ and $k$ is studied. For a fixed orientable surface $S_k$ of genus $k$, let $A_{xy}^k$ be the minor-closed class of graphs with terminals $x$ and $y$ that either embed into $S_{k-1}$ or admit an embedding $\Pi$ into $S_k$ such that there is a $\Pi$-face where $x$ and $y$ appear twice in the alternating order. In this paper, the obstructions for the classes $A_{xy}^k$ are studied. In particular, the complete list of obstructions for $A_{xy}^1$ is presented.