  Let G be a graph with vertices V and edges E. Let F be the union-closed
family of sets generated by E. Then F is the family of subsets of V without
isolated points. Theorem: There is an edge e belongs to E such that |{U belongs
to F | e belongs to U}| =< 1/2|F|. This is equivalent to the following
assertion: If H is a union-closed family generated by a family of sets of
maximum degree two, then there is an $x$ such that |{U belongs to H | x belongs
to U}| > 1/2|H|. This is a special case of the union-closed sets conjecture. To
put this result in perspective, a brief overview of research on the
union-closed sets conjecture is given. A proof of a strong version of the
theorem on graph-generated families of sets is presented. This proof depends on
an analysis of the local properties of F and an application of Kleitman's
lemma. Much of the proof applies to arbitrary union-closed families and can be
used to obtain bounds on |{U belongs to F | e belongs to U}|/|F|.
