Let $G=(V,E)$ be a connected, undirected graph. Define the boundary $\partial S$ of a set $S\subset V$ to be the set of all $v\notin S$ joined to $S$ by an edge, i.e. $$\partial S = \{v\not in S: \exists v'\in S s.t. {v,v'}\in E\}.$$ We know that, if there are at least $n$ vertices of degree $\geq 3$ in $G$, then there is a set $S\subset V$ such that the subgraph $G|_{S}$ is connected and $\partial S$ has $\geq n/4+2$ elements.

(Proof: construct a spanning tree with $\geq n/4+2$ leaves, as per Spanning trees: the last darn $1/4$ . Then take $S$ to be the set of internal nodes. Thanks to F. Petrov (Existence of connected component with large boundary?).)

Let $V'\subset V$ be set of $m$ vertices, all of degree $\geq 3$. Is there a set $S\subset V$ such that $G|_S$ is connected and $\partial S\cap V'$ has $\geq \delta m$ elements, where $\delta=1/100$, say?

Let $G=(V,E)$ be a connected, undirected graph. Define the boundary $\partial S$ of a set $S\subset V$ to be the set of all $v\notin S$ joined to $S$ by an edge, i.e. $$\partial S = \{v\not \in S: \exists v'\in S s.t. {v,v'}\in E\}.$$ We know that, if there are at least $n$ vertices of degree $\geq 3$ in $G$, then there is a set $S\subset V$ such that the subgraph $G|_{S}$ is connected and $\partial S$ has $\geq \frac{n}{4}+2$ elements.

(Proof: construct a spanning tree with $\geq \frac{n}{4}+2$ leaves, as per Spanning trees: the last darn $1/4$ . Then take $S$ to be the set of internal nodes. Thanks to F. Petrov (Existence of connected component with large boundary?).)

Let $V'\subset V$ be set of $m$ vertices, all of degree $\geq 3$. Is there a set $S\subset V$ such that $G|_S$ is connected and $\partial S\cap V'$ has $\geq \delta m$ elements, where $\delta=1/100$, say?