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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 $S_k$ ofgenus $k$, 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 thealternating order. In this paper, the obstructions for the classes $A_{xy}^k$are studied. In particular, the complete list of obstructions for $A_{xy}^1$ ispresented.

Obstructions for two-vertex alternating embeddings of graphs in surfaces