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Considering loop free cubic graphs (graphs where every node has 3 neighboring nodes): Is is possible to construct a spanning tree that only has nodes with 3 neighbors in the spanning tree or 1 neighbor in the spanning tree (leaves).

That is I want to be able to construct a spanning tree where there are no nodes that are connected to only 2 other nodes in the spanning tree. They should all be connected to either 1 node (a leaf) or all their edges in the underlying cubic graph should also be present in the spanning tree (ie attached to 3 nodes in the spanning tree)?

Does anyone know the answer to this? And if it is possible, how to construct such a spanning tree? Many thanks.

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I'm trying to search for similar questions/results (particularly this question restricted to cubic polyhedral graphs) but having trouble finding appropriate search terms. Is there a standard name for this kind of tree? "trivalent tree" seems like it would be a natural name for it, and seems to confirm it, but unfortunately seems to contradict it (allowing vertices of degree 2). Confusing. – Don Hatch Jun 26 at 14:05

It's not always possible. For example:


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