Q:

Edges per vertex in polyhedron

Which polyhedra have no more than 3 edges per vertex?
limitations:  No more than 20 faces.  Convex polyhedra only.  No limit to the number of vertices.  No limit to the number of edges, only to the number of edges-per-vertex.  The polyhedra need not be regular.  
I know Tetrahedron and hexahedron (cube, and variants) qualify.  Are there any others and, if so, what are they?

A:

There are a lot.
Tetrahedron, Cube, Dodecahedron, Duerer Solid, all prisms with number of sides less than 18.
Then you can take any of above and truncate any number of vertices (while number of faces does not exceed 20). If you haven't reached 20 faces, you can repeat truncation.
Then you can take a polyhedron with 4 edges at vertex and if it has faces+vertices<20, truncation of all vertices will get you a proper polyhedron like Truncated Octrahedron.

