Is there a set $S$ of $O(n^{1-\alpha})$ vertices in any regular undirected graph $G$ on $n$ vertices with degree $\Omega(n^\alpha)$ where $\alpha\in(0,1)$ such that every edge in the graph is incident on at least one vertex in $S$?

Moreover consider the subgraph $G_S$ with vertex set $S$ and edge set consisting of edges in $G$ with both end points in $S$. Can we pick an $S$ such that subgraph $G_S$ is regular? What is the least degree that $G_S$ can have?