Let $d = (d_1,d_2,..,d_n)$ be a graphical sequence, i.e., a degree sequence from which one can construct a graph $G$ with $n$ vertices and $2^{-1}\sum_{i=1}^n d_i$ edges.
Consider for $i,j\in\{1,...,n\}$ the transformation $\mathcal{T}_{ij}:\{1,...,n-1\}^n\to\{0,...,n-1\}^n$ such that $\mathcal{T}_{ij} (d_1,...,d_n) = (d_1,...,d_i+1,...,d_j-1,...,d_n)$, whenever $d_i < n-1$ otherweise $\mathcal{T}_{ij} (d_1,...,d_n)=(d_1,...,d_n)$.
Are there non-trivial conditions on $(d_1,d_2,..,d_n)$ such that $\mathcal{T}_{ij} (d_1,...,d_n)$ is a graphical sequence (possible 0 entries representing vertices without edges)? Evidently, this does not hold in general. Consider for example the sequence $(3,3,2,2)$ which is graphical but $\mathcal{T}_{34}(3,3,2,2)=(3,3,3,1)$ is not graphical.
I would in general be interested in transformations which preserve the property of a degree sequence to be graphical.