TITLE: Extremal examples for a folklore lemma on subgraphs of large minimum degree
QUESTION [7 upvotes]: It's a well known fact that a graph $G$ of average degree $d$ has a subgraph $G'$ of minimum degree at least $d/2$ and that the constant $1/2$ cannot be improved. The proof I know, which proceeds by peeling away low degree vertices, doesn't seem to characterise, in any nice way, the extremal examples for which the constant $1/2$ cannot be improved.
So my question is, what do extremal examples "look like"? Is there a simple characterisation? If not, I'd love to see any interesting constructions/families which give infinitely many extremal examples for a given value of $d$.

REPLY [4 votes]: I believe that David Eppstein's construction is basically the whole story. 
So for $0 <\epsilon \leq 1/2 $, call a graph 'bad' if it fails to have a subgraph with min degree $> (1/2 +\epsilon)d$.
Now Eppstein's construction restated:
" Build a graph on $1,...,n$ by visiting each vertex $i= 1,2.. $ and, at each stage, join $i$ to $(1/2 + \epsilon(i))d $ additional vertices among $\{ i+1, \ldots ,n \} $" 
Where $|\epsilon(i)| \leq \epsilon$ and $\sum \epsilon(i) = 0$. One thinks of this $\epsilon(i)$ as the 'noise' allowed by the $\epsilon$-room that we have. 
The average vertex has $d/2$ neighbors ahead of it and was hit 
by $d/2$ edges from previous vertices so the average degree is $d$. Of course, there are some issues at the boundary but this does not affect the average if we take n large.
Now we show that every 'bad' graph is constructed in this way. Obverse that bad graphs are the ones that are reduced to the empty graph by the process:
"If there is a vertex of degree $\leq (1/2 + \epsilon)d$ remove it. Otherwise, STOP".
Thus for a bad graph,we get an ordering of the vertices $v_1,v_2,...,v_n$ (taking in the order removed by the process) so that 
$v_i$ has at most $(1/2 + \epsilon)d$ edges forward to $v_{i+1}, \ldots v_n$. If the number of forward edges of $v_i$ is $(1/2 + \epsilon(i))d$ then $\sum \epsilon(i)=0$ holds by the fact the average degree is $d$.