Finding a triangle graph given a list of point degrees

Question

I have a theorem: Given a list of points on a 2 dimensional plane, and the degree of each point, there should correspond only one way (counting isomorphic graphs as the same) 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?(probably NP-hard if so)

example of type of graph

Answer 1

Here is a counterexample: two triangular meshes with the same degree sequence, but different edges drawn.

(The outer face isn't a triangle, but neither is the outer face in your example, and in any case that's easily fixed by drawing an extra structure around both meshes.)