TITLE: Smallest Connected Graph for Given Degree Sequence
QUESTION [8 upvotes]: For a given integer sequence $(d_1, d_2,...,d_n)$, a natural question is if such a sequence is  graphical, i.e. is a degree sequence of some graph. According to Erdős–Gallai theorem, A sequence of non-negative integers $d_1\geq\cdots\geq d_n$ can be represented as the degree sequence of a finite simple graph on $n$ vertices if and only if $d_1+\cdots+d_n$ is even and
$\sum^{k}_{i=1}d_i\leq k(k-1)+ \sum^n_{i=k+1} \min(d_i,k)$
holds for $1\leq k\leq n.$
My questions are 
(1) If Erdős–Gallai theorem holds, what is the condition that this graph is unique? 
(2) If those graphs are not unique, how to find a connected graph with smallest connectivity among them?

REPLY [8 votes]: A theorem of Hakimi says
that any pair of degree-equivalent graphs can be obtained one
from the other by a sequence of "elementary $2$-switchings"
(probably known under many other names), which involve
the subgraph switch on the subgraph induced
by four vertices, as illustrated in one instance below.

 


So whatever you seek to minimize (cf. the comments),
likely it could be pursued by searching for the minimum
via these $2$-switchings.


Hakimi, S. Louis. "On realizability of a set of integers as degrees of the vertices of a linear graph. I." Journal of the Society for Industrial & Applied Mathematics. 10.3 (1962): 496-506.
Hakimi, S. Louis. "On realizability of a set of integers as degrees of the vertices of a linear graph II. Uniqueness." Journal of the Society for Industrial & Applied Mathematics. 11.1 (1963): 135-147.