Non-overlapping [network] clusters
The picture shows all distinct clusters with 3 dangling connections and 9 nodes that are not self-overlapping. The only smaller cluster with the same property is the trivial one with just a single node.
Most clusters that can overlap will be able to do so in an infinite number of possible networks. (One can see this by noting that they can overlap inside clusters with dangling connections, not just closed networks.) But there are some clusters that can overlap only in a few small networks. The pictures below show examples where this happens. The pictures in the main text still treat such clusters as non-overlapping.
If two clusters overlap, then this means that there is some network in which there are copies of these clusters that involve some of the same nodes. And it is possible to search for such a network by starting from a single node and then sequentially trying to take corresponding pieces from the two clusters.