# graph on mobius strip is nonplanar? [duplicate]

Consider the closed curve on the mobius strip that covers both sides of the fundamental polygon: https://thumb9.shutterstock.com/display_pic_with_logo/1663882/462955276/stock-vector-blue-moebius-strip-or-moebius-band-with-centerline-surface-with-only-one-side-and-one-boundary-462955276.jpg

Is this curve viewed as a graph (with one vertex?) non planar since it must intersect itself in the plane? (help me make this more rigorous)

How do I more rigorously prove that certain graph in the mobius strip is non planar? Do I need to use some planarity criterion like Kuratowski's?