  We discuss, on general grounds, how two subgraphs of a given Feynman graph
can overlap with each other. For this, we use the notion of connecting and
returning lines that describe how any subgraph is inserted within the original
graph. This, in turn, allows us to derive "non-overlap" theorems for
one-particle-irreducible subgraphs with $2$, $3$ and $4$ external legs. As an
application, we provide a simple justification of the skeleton expansion for
vertex functions with more than five legs, in the case of scalar field
theories. We also discuss how the skeleton expansion can be extended to other
classes of graphs.
