The New York Times online Games section has a daily puzzle called

Vertex
The player is faced with an array of numbered dots and connects them with segments.

How to play

- Draw lines between points to create triangles.
- Connect vertices to create triangles and assemble an image.
- The number on a vertex shows its remaining connections.
- Triangles will fill in if they are correct.
- Double tap a vertex to clear its connections.

I should add that the finished puzzle is a connected network of triangles and every segment is part of a triangle.

Question: is a solvable Vertex puzzle guaranteed to have a unique solution?

(In reply to

re: Does this count? by Jer)

I did spend some time on the Vertex site and came across a puzzle that did have one triangle connected by just a single vertex. When solved the figure was an apple or some other fruit, I think. But that seems to be an outlier with the expectation that all the triangles are connected edge-wise, too.

Adding the extra constraint that each triangle must share an edge with another triangle I concocted this:

A---------B

| \ / |

| C---D |

| | E | |

| F---G |

| / \ |

H---------I

It is hard to draw in ASCII, so this is just half-solved. Vertices A, B, E, H, and I will have four total connections and Vertices C, D, F, and G will have five total connections.

Draw in all of CE, DE, FE, GE to split square CDFG into four triangles.

Then two solutions come from two ways to split all the surrounding trapezoids into two triangles: Draw AD, BG, IF, and HC; or draw AF, BC, ID, HG.