Sparse subgraph preserving rooted edge connectivity up to $k$
Let $d^+_G(x)$ be the in-degree of $x$ in graph $G$.

Theorem (Lovász 1973): 
For a directed graph $G$ and a specified vertex $r$, there exist a subgraph $G'$ with the property that $d^+_{G'}(x) = \lambda(r,x,G') = \lambda(r,x,G)$.

The desired sparse graph exists, as we can keep removing edges to reach a minimal graph with the desired connectivity property.