I am looking for the class of edge-maximal graphs without a K3 minor. I know that a graph is called edge-maximal for a property if one cannot add another edge without loosing the property. I also know that a graph that is not allowed to contain a K3 minor has to be cycle-free and thus has to be a tree because when adding one edge to a tree it is not possible to keep the graph cycle-free. Is this enough to find the class of edge-maximal graphs without a K3 minor?
2026-03-25 11:34:55.1774438495
Class of edge-maximal graphs without a K3 minor
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As you already stated, any graph $G$ without a $K_3$ minor such that adding any new edge yields a graph with a $K_3$ minor must be a tree, so your graph class is contained in the class of trees. For the reverse direction note that any tree, i.e a connected graph without cycles (equivalent to the existence of a $K_3$ minor), is also in your class. We conclude that your class is indeed the class of trees.