All minimal non-apex graphs with 12 or fewer vertices or 26 or fewer edges are listed, and every 13-vertex graph with minimum degree 6 is shown to be apex or to contain a K6 minor.
On a spatial analogue of Kuratowski’s theorem on planar graphs - an open problem.Graph Theory, eds
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New minor minimal non-apex graphs
All minimal non-apex graphs with 12 or fewer vertices or 26 or fewer edges are listed, and every 13-vertex graph with minimum degree 6 is shown to be apex or to contain a K6 minor.