Suppose we have a finite connected graph $G$, I want to add 2 cells to $G$ so that the 2 cells have boundaries of length 4 (squares) and so that $G$ is the 1 skeleton of a surface (2manifold) without boundary. To check that such a process can be done to $G$ does it suffice to check whether each edge in $G$ occurs as an edge of 2 distinct 4 cycles? does it suffice to check if each edge occurs as exactly 2 distinct 4 cycles? Is there a known sufficient condition?
