Interactive Tree Decomposition

Here's a tool (in Mathematica) to help visualize the process of constructing a Junction Tree. wrong link fixed

Clicking on vertices corresponds to vertex elimination and each click creates a new bag. After all vertices are eliminated, junction tree is created by removing redundant bags and taking the maximum spanning tree with size of the separators defining the weight of each edge as the junction tree.

If you try it on small examples, you can get some intuition of how outperform the greedy approach -- basically you want to find small separators of the graph, and at the same time you want those separators to have a lot of vertices in common