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.
Searching for and Classifying the Finite Set of Minor-Minimal Non-Apex Graphs
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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.