(Please click for bigger - there is quite a bit of detail :)
The left panel shows the search tree of structures; each circle is a structure, the red ones at the tips of some branches are fully connected and saturated. The blue branch has been selected, and is shown in the middle, with the structure at the tip at the top.
This solution is also highlighted (in red - colors are not very consistent) and shown in detail on the right. The right/middle panel is a traditional molecule layout, but with numbers in place of atom symbols. The lower right panel is the spanning tree of the atom highlighted in red in the upper right panel.
Clearly, there are still a lot of unnecessary structures generated since most branches are dead ends. However, this version does at least avoid duplicates - sadly it also misses a few structures :( Clearly refining partitions based on element symbols doesn't totally work...