finding edges of a triangle graph from degrees of points

Question

my theory: Given a list of points on a 2 dimensional plane, and the degree of each point, there should correspond only one way to arrange the edges between points so that the final graph is a mesh of only triangles, and the degree of every point is correct.

MY question is, is my theory true, and if so, if there is a way to find the unique graph given just the degrees of each point?(might be very difficult)

I know my theory isn't very formal, I'll try to clarify as much as possible if needed. I wasn't sure of the correct terminology, but what I mean by a triangle mesh is something like this example of graph

Answer 1

Assuming I've understood the question correctly, here's a counterexample: