Submultisets and Graphs

The previous post mentioned the restricted weak composition (RWC), but didn't expand on it at all! Basically, I found this excellent paper : "Generalized Algorithm for
Restricted Weak Composition Generation" by Daniel R. Page. It even gave some java code in an appendix - good stuff :)

Anyway, a RWC is a composition (which is a partition where order matters) that is weak - has zeros in it - and the parts are restricted. So [1, 0, 1, 1] is a weak composition of 3 into 4 parts and lets say we have restricted the parts to {0, 1, 2}. Here is an overview of the scheme:

where we take a degree sequence, convert to a multiset and use a RWC to get a particular sub-multiset. This allows us to take some count of some subset of elements from the multiset.

Doing this for all sub-multisets at each round should then allow us to list graphs - although not without redundant examples:

 This shows all starting points for 3 -> [3, 2, 2, 1, 1] and the eventual graphs that are formed. Note that a) and e) are isomorphic. Still, this would surely remove a lot of redundant solutions.