Let $\Gamma = (V,E)$ be a connected (undirected) graph where every vertex has degree $\geq 2$. Let $E'\supset E$ be a larger set of edges between elements of $V$ such that every vertex of $\Gamma'=(V,E')$ has degree $\geq 4$.
Must there exist a subset $V'\subset V$ such that $\Gamma'|_{V'}$ is connected and $V'$ has large boundary in $\Gamma$? ("Large boundary" here means: $\geq \epsilon |V|$ elements of $V\setminus V'$ are connected by an edge to $V'$) If not, what are some counterexamples?
(For $E=E'$, the problem is not difficult (even if we just assume that many vertices of $\Gamma'$ have degree $\geq 3$, rather than that they all have degree $\geq 4$. See Existence of connected component with large boundary?)
