*[Posted this first at math stackexchange, but it probably fits better here.]*

I am looking for references about the following problem.

Given a (connected) bipartite graph $G$, find an independent set $M$ minimizing the ratio $|\partial M|/|M|$.

Here, $\partial M$ denotes the set of neighbours of $M$.

An independent set is a set $M$ of vertices such that $(\partial M)\cap M = \emptyset$.

More specifically, I am looking for information about the class of bipartite ~~graphs~~ *trees* $G= (A\,\dot{\cup}\, B \,, E)$ for which $A$ is *the unique solution* to the problem above.

Examples of such graphs include lines with an odd number of vertices, or, more generally, trees in which every $b\in B$ has degree two.

2more comments