This question is inspired by this challenge and this paper.
In the paper it is claimed that it is possible to use the algorithm so that the resulting graph will always be connected if a connected realization exists.
I'm very interested, is it possible somehow to include multi-edge realization (keeping connectivity)?
Obviously, we can subtract each iteration not 1, but other numbers, but how to choose them correctly?
Here is the answer:
Theorem 7 (connected loopless multigraph construction).
Let $d$ be a multigraphical degree sequence, and let us repeatedly select the largest remaining degree vertex and the smallest non-zero remaining degree vertex, and connect them with a single edge. This construction procedure results in a connected graph if and only if $d$ was potentially connected.