Abstract: A graph in a manifold Mⁿ is a union of smooth arcs, which begin and end in two of the vertices of the graph. The number of arcs meeting at a vertex is called the degree of the vertex. In a physical experiment, it is possible to dip a wire framework into soap solution and withdraw it: a surface may be formed in which three pieces of soap film meet with equal angles along edges, and where six pieces of soap film may meet symmetrically at a point. Jean Taylor in 1976 used a certain class of rectifiable sets in R³, which she called soap film-like surfaces, as a model for such physical surfaces, and proved their interior regularity. The existence had been established earlier by Fred Almgren. We extend recent work of Choe and the speaker on branched minimal surfaces to this case (with an extra assumption of boundary regularity), proving that the density of a soap film-like surface may be controlled by the total curvature of its boundary graph. The contribution to the total curvature at a vertex is defined as the supremum over p of the sum of the complements of the interior angles to a cone over the graph with vertex p in Rⁿ. This is joint work with Sumio Yamada.