Graphs of vector valued maps: decomposition of the boundary

We consider non-smooth vector valued maps such that the current carried by the graph has finite mass. We give a suitable decomposition of the boundary of the graph-current, provided that it has finite mass, too. Every such a component is a nice current whose support projects on a subset of the domain space that has integer dimension. This structure property is a consequence of a more general one that is proved for ``vertical" integer multiplicity rectifiable currents satisfying a null-boundary condition. As a consequence, for wide classes of Sobolev maps such that the graph is a normal current, we shall prove that the singular part of the distributional minors of order k is concentrated on a countably rectifiable set of codimension k, hence no ``Cantor-type" part appears. The corresponding class of functions of bounded higher variation is studied, too. Finally, we discuss a possible notion of singular set in our framework, and illustrate several examples.