# NP completeness of classes of spanning trees

I am teaching a complexity course, and I want to give some examples of similar looking problems such that one is in P, and the other is NP complete.

This made me think of the following problem: does a given graph contain a spanning tree with a given degree sequence? This is obviously easy for the sequence $n-1,1,...,1$ and hard for $2,...,2,1,1$. Is there a general answer?

Lemke has shown that it is NP-hard to decide whether a cubic graph on $n=2k$ vertices has a spanning tree in which $k+1$ vertices have degree $1$ and $k-1$ vertices have degree $3$.
• Lemke's result implies the NP-completeness of deciding whether there exists a spanning tree without vertices of degree $2$.
• A slight modification of Lemke's reduction shows that the following problem is NP-complete for every fixed integer $p\ge4$: Decide whether there exists a spanning tree, in which every non-leaf vertex has degree at least $p$.
• The same modification shows that the following problem is NP-complete for every fixed integer $p\ge4$: Decide whether there exists a spanning tree, in which every vertex degree lies in $\{1,p\}$.