Abstract
Abstract We show that every 5-connected graph admitting an embedding into the projective plane with face-width at least 3 contains K 6 as a minor. Further we find a simple proof that every 5-connected planar graph contracts to icosahedron. We adapt the proof and find a simple face-width type condition such that every 5-connected graph embedded in an arbitrary surface satisfying the condition contracts to icosahedron.
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