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 Sk of genus k, let be the minor-closed class of graphs with terminals c07d1a36c80750fe4c4c2d983e3d87" title="Click to view the MathML source">x and 20c43011bd1b2ca6791a0ccac5f" title="Click to view the MathML source">y that either embed into Sk−1 or admit an embedding Π into Sk such that there is a Π-face where c07d1a36c80750fe4c4c2d983e3d87" title="Click to view the MathML source">x and 20c43011bd1b2ca6791a0ccac5f" title="Click to view the MathML source">y appear twice in the alternating order. In this paper, the obstructions for the classes are studied. In particular, the complete list of obstructions for is presented.