# Given a free involution $i$ on a finite graph $G$, is there a minimal embedding of $G$ such that $i$ is facial?

This question comes from my own Bachelor-thesis work. I am exploring the 1-2-inifity conjecture, or Negamis conjecture, and I am trying to see what happens if one tried to extend Negamis result that a connected graph has a 2-fold planar cover iff it is embeddable in the projective plane to higher genus surfaces.

The question in itself is only indirectly linked here, but I find it interesting in its own right. To me it looks like something that should be obviously true, but I have stared at it for a little bit too much time to really know. And I just cannot figure it out. Frankly, I'm not sure whether this is an open question or not.