Let $X$ be a (locally finite) metric graph (all of whose edges are length 1). A subset $A \subset X$ is contracting if there exists a constant $C \geq 0$ such that the nearest point projection on $A$ of any ball disjoint from $A$ has diameter at most $C$.
Question: Let $A,B \subset X$ be two subsets at finite Hausdorff distance. If $A$ is contracting, is $B$ necessarily contracting?
I proved the answer is yes if $A$ is the $\epsilon$-neighborhood of $B$, ie. the set of points at distance at most $\epsilon$ from $B$. More generally, my first idea was to suppose by contradiction that $B$ is not contracting, and then to look at what happens in a well-chosen asymptotic cone, but I was not able to conclude.
