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Vertices are points in a solid, or 3-D figure, where edges meet.
There is no 3 dimensional figure with JUST 3 edges.
vertex
A parallelopiped, which is the 3-dimensional equivalent of a parallelogram.
None. Using Euler's formula v - e + f = 2, where v is vertices, e is edges, and f is faces, we see that for your question f = 3. No solid figure can have less than 4 faces (a tetrahedron).