Let $G=(V,E)$ be a simple undirected graph. Define an *mmd$k$s in $G$* (for 'maximal minimum degree $k$ subset') to be any subset $S$ of $V$ such that 


1. the subgraph induced by $S$ in $G$ has minimum degree $\geq k$,
2. $S$ is $\subseteq$-maximal w.r.t. property 1.


Moreover, for any $k$, let $\mathrm{smmds}(G,k):=\sup\{ \lvert S \rvert \colon\text{$S$ is an mmd$k$s in $G$}\}$.


Moreover, for any class $\mathbb{G}$ of graphs, let $\mathrm{smmds}(\mathbb{G},k):=\sup\{ \mathrm{smmds}(G,k)\colon G\in\mathbb{G}\}$.


My question is whether 


* *mmd$k$s*'s
* the graph invariant $\mathrm{smmds}(\cdot,k)$,


have already been analysed and named in graph theory.


I am interested in understanding how $\mathrm{smmds}(\cdot,k)$, varies, as a function of $k$, for different families of graphs (or certain random graph models). 



Subsets of a graph, maximal w.r.t. the property of inducing a subgraph with minimum degree at least $k$
According to [Wikipedia](https://en.wikipedia.org/wiki/Degeneracy_(graph_theory)#k-Cores), a *$k$-core* of a graph $G$ is a maximal connected subgraph $K$ of $G$ such that every vertex of $K$ has degree at least $k$. Apart from the "connected" part of the definition, this is the same as your set $S\_k$. Your set $S\_k$ is the union of the vertex sets of all $k$-cores of $G$. 


Similar to what Gerhard suggested, you can find the set $S\_k$ by initialising $S\_k:=V(G)$ and then repeatedly deleting every vertex of degree less than $k$ in the subgraph induced by $S\_k$ until there are no such vertices left (note that this may terminate with $S\_k=\emptyset$). 


With regards to $k$-cores in random graphs, one natural place to start might be the paper [*Size and connectivity of the $k$-core of a random graph*](http://www.sciencedirect.com/science/article/pii/0012365X9190162U) by Tomasz Łuczak.


