Let $\mathscr{C}$ be a graph consisting of an $n$-cycle, i.e., $n$ vertices arranged in a circle, with undirected (or two-way) blue edges between neighbors. Let us now draw two additional, green arrows from each vertex $n$ to two vertices $m$, $m+1$ that are not its neighbors. Denote by $\mathscr{C}'$ the new graph, including all edges, blue and green.

Must there exist a subset $S$ of the set of vertices $\{1,2,\dotsc,n\}$ such that $S$ is connected in $\mathscr{C}'$ and have large boundary in $\mathscr{C}$? By "large", I mean "having more than $\delta n$ elements for some fixed $\delta>0$".

(If not, can you give a counterexample? Can counterexamples be easily classified?)

Let $\mathscr{C}$ be a graph consisting of an $n$-cycle, i.e., $n$ vertices arranged in a circle, with undirected (or two-way) blue edges between neighbors. Let us now draw two additional, green arrows from each vertex $n$ to two vertices $m$, $m+1$ that are not its neighbors. Denote by $\mathscr{C}'$ the new graph, including all edges, blue and green.

Must there exist a subset $S$ of the set of vertices $\{1,2,\dotsc,n\}$ such that $S$ is connected in $\mathscr{C}'$ and have large boundary in $\mathscr{C}$? By "large", I mean "having more than $\delta n$ elements for some fixed $\delta>0$".

(If not, can you give a counterexample? Can counterexamples be easily classified?)

EDIT: Aha. What if the green arrows are not arrows but undirected edges (or two-way arrows, if you wish)?